arXiv:cond-mat/0206564v1 28 Jun 2002
The asymmetric lossy near-perfect lens
S. Anantha Ramakrishna
∗
, J.B. Pendry
†
The Blackett laboratory, Imperial College, London SW7 2BZ, UK
D. Schurig, D.R. Smith
‡
and S. Schultz
Department of Physics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0319, USA
We extend the ideas of the recently proposed perfect lens [J.B. Pendry, Phys. Rev. Lett. 85,
3966 (2000)] to an alternative structure. We show that a slab of a medium with negative refractive
index bounded by media of different positive refractive index also amplifies evanescent waves and
can act as a near-perfect lens. We examine the role of the surface states in the amplification of the
evanescent waves. The image resolution obtained by this asymmetric lens is more robust against
the effects of absorption in the lens. In particular, we study the case of a slab of silver, which has
negative dielectric constant, with air on one side and other media such as glass or GaAs on t he other
side as an ‘asymmetric’ lossy near-perfect lens for P-polarized waves. It is found that retardation
has an ad verse effect on the imaging due to the positive magnetic permeability of silver, but we
conclude that subwavelength image resolution is possible inspite of it.
I. INTRODUCTION
The electromagnetic radiation emitted or sca ttered by an object consists of a radiative component of propagating
modes and a near-field compo ne nt of non-propagating modes whose amplitudes decay exponentially with distance
from the sourc e. For a mo nochromatic s ource, the electromagne tic field in fr ee space can be expressed as a Fourier
sum over all wave vectors:
E(x, y, z; t) =
X
k
x
,k
y
,k
z
E(k
x
, k
y
, k
z
) exp[i(k
x
x + k
y
y + k
z
z − ωt)] (1)
where k
2
x
+ k
2
y
+ k
2
z
= ω
2
/c
2
and c is the sp eed of light in free space . For k
2
x
+ k
2
y
> ω
2
/c
2
, k
z
is seen to be purely
imaginary, and the case is similiar for k
x
and k
y
. The near-field consists of these partial waves with imag inary wave
vectors and decays exponentially away from the source. These are the high-frequency Fourier components describing
the fast varying s patial features on the object and are never detected in conventional imaging using conventional
lenses. This la ck of information results in the limitation to conventional imaging that sub-wavelength features of a
source cannot be resolved in the image. In order to over come this limitation, scanning near-field optical microscopy
(SNOM) wa s proposed where the near-field of the radiating object is probed by bring ing a tapered fibre tip very close
to the object (See Ref. [1] for a recent rev iew). A rule of the thumb is that the near-field produced by a (periodic)
feature of spatial extent d will decay exponentially at a rate of d/2π away from the surface. Hence there is a need to
get close to the surface at these resolutions, just in order to detect the evanescent field. Impre ssive advances have been
made in this field, but there still remain several problems in understanding the images, phase contrast mechanisms
and a ssociated artifacts.
Now in order to completely reconstruc t the object, we would need to amplify these evanescent waves a nd give the
appropriate phase shift for the propagating components. This is precisely what the recently proposed perfect lens
with negative refractive index accomplishes [2]. Vese lago had noted a long time ago [3] that due to the reversal of
Snell’s law, a slab of negative refractive index would act as a lens in that the rays from a source on one side would
get refocussed on the other side to form an image. But the amplification o f evanescent waves by a slab of negative
refractive index no ted by Pendry [2] was a surprising and ne w result. The possibility of negative refractive media ha s
already been experimentally demonstrated in the microwave region of the spectrum [4,5]. Both the dielectric constant
and the magnetic permeability are negative in a negative refractive medium, and the electromagnetic waves in such
a medium will be left-handed as a consequence of Maxwell’s equations. No known natural materials have negative
magnetic permeability, and the negative refractive media are meta-materials consisting of interlaced periodic arrays
∗
E-mail: s.a.ramakrishna@ic.ac.uk, Tel: +44 020 75947597, FAX: +44 020 75947604
†
E-mail: j.pendry@ic.ac.uk
‡
E-mail: drs@ucsd.edu
1
of split ring resonators [6] and thin wires [7]. Pendry also showed that a thin slab o f silver with negative die le ctric
constant would act a s near-perfect lens for near-field imaging with P-polariz e d waves in the electro-static limit, with
the res olution limited only by absorption in the lens (silver). More recently, we have examined some consequence s of
deviations in the dielectric constant and magnetic permeability from the perfect lens conditions (ǫ = −1, µ = −1) on
the res olution of the lens [8] and found that the restrictions on ǫ, µ were quite severe, but achieveable by current day
technology. The main advantage of this near-field perfect lens over the current methods of SNOM is that a complete
image is generated a t the image plane. This would be important in many applicatio ns, for example, in-situ imag ing
of biological molecules and processes by flourescence imaging.
Here we extend the ideas of Pendry’s original work. We show that an asymmetr ic lens consisting of a slab of
negative refractive index bounded by media of different positive refractive index can a lso act as a near-perfect lens.
We then examine the nature of the surface modes which are responsible for the amplification of the evanesce nt waves.
In particular, we study the case of a film of silver (with negative dielectric constant) deposited on other media such
as glass or GaAs with positive dielectric constant as a near-field imaging lens at optical frequencies. We show that
the as ymmetry in the system can actually enhance the resolution, depending on the choice of asymmetry and the
operating frequency. One of the main advantages of this asymmetric lens is that a very thin metal film deposited on
a solid substr ate (Glass or GaAs) will be mechanically much more stable than a free-standing metal film. We also
study the effects of retardation on the system and conclude that sub-wavelength resolution is possible inspite of the
adverse effects of retardation and absorptio n.
II. THE ASYMMETRIC SLAB
[Insert figure 1 about here ]
Typically we consider the asymmetric lens to be a slab of negative re fractive medium of thickness d, dielectric
constant ǫ
2
and ma gnetic permeability (µ
2
) be tween media of differing refractive indices (See figure 1). We will
consider the medium-1 on one side of the s ource to be air (ǫ
1
= 1, µ
1
= 1) and some other die lectric or magnetic
material (ǫ
3
, µ
3
) on the other side. We will consider the object plane to be in medium-1 at a distance d/2 and the
image plane, where we detect the image inside medium-3 at a distance d/2 from the edge of the slab. Let P-polar ized
light be incident on the slab from the medium-1 , with the ma gnetic field
1
given by
H
1p
= exp(ik
(1)
z
z + ik
x
x − iωt), (2)
where k
2
x
+ k
(j)
z
2
= ǫ
j
µω
2
/c
2
(j = 1, 2, 3 for the different media ). We will work in two dimensions for reasons
of simplicity. If the index of refraction is negative (ǫ
2
< 0, µ
2
< 0), then the Maxwells e quations and causality
demand that k
(2)
z
= −
p
ǫ
2
µ
2
ω
2
/c
2
− k
2
x
if k
x
<
√
ǫ
2
µ
2
ω/c, and k
(2)
z
= i
p
k
2
x
− ǫ
2
µ
2
ω
2
/c
2
if k
x
>
√
ǫ
2
µ
2
ω/c [2]. The
transmission coefficient for the P-polarized light across the slab is
T
p
(k
x
) =
4 (
k
(1)
z
ǫ
1
) (
k
(2)
z
ǫ
2
) exp(ik
(2)
z
d)
(
k
(1)
z
ǫ
1
+
k
(2)
z
ǫ
2
)(
k
(2)
z
ǫ
2
+
k
(3)
z
ǫ
3
) − (
k
(1)
z
ǫ
1
−
k
(2)
z
ǫ
2
)(
k
(3)
z
ǫ
3
−
k
(2)
z
ǫ
2
) exp(2i k
(2)
z
d)
. (3)
We immediately see that if either ǫ
2
= −ǫ
1
and µ
2
= −µ
1
, or ǫ
2
= −ǫ
3
and µ
2
= −µ
3
, then T
p
= [2ǫ
3
k
(1)
z
/(ǫ
1
k
(3)
z
+
ǫ
3
k
(1)
z
)] exp(−i k
(2)
z
d) and the amplification of the evanescent waves as well as the phase reversal for the propagating
components results. Thus, the system does not have to be symmetric (ǫ
1
= ǫ
3
, µ
1
= µ
3
) as in the original work of
Pendry for accomplishing amplification of evanesc ent waves. In this asymmetric case, however, the field str ength at
the image plane differs from the object plane by a constant factor, and thus the imag e intensity is changed. Further,
it is eas ily verified that wave-vectors with differe nt k
x
will refocus at slightly different positions unless ǫ
3
= ǫ
1
and
µ
3
= µ
1
(i.e. the symmetric case) or in the limit k
x
→ ∞. Thus there is no unique (perfect) image plane and we
should ex pect the image to suffer from abe rrations. Hence, we term the asymmetric slab a near-perfect lens. A similiar
result holds for the S-polarized wave incident on the slab. Again, we point out that with this asymmetric lens, the
1
It is more convenient to use the magnetic field for the P-polarized light . The electric field can be obtained by using the
Maxwell’s equation
~
k
(j)
×
~
B = −ωǫ
j
~
E.
2
object is considered to be in medium-1 (usually air) and the image is formed inside medium-3 (mo stly considered to
be a high index dielectric in subsequent sections of this paper).
In the electrostatic (magnetostatic) approximation (k
x
→ ∞), we have k
(j)
z
→ ik
x
and the dependence of the
transmission coefficient for the P(S)-polarize d field on µ(ǫ) is eliminated. The transmissio n coefficient for the P-
polarized wave across the s lab is then given by
T
p
(k
x
) =
4ǫ
2
ǫ
3
exp(−k
x
d)
(ǫ
1
+ ǫ
2
)(ǫ
2
+ ǫ
3
) − (ǫ
2
− ǫ
1
)(ǫ
2
− ǫ
3
) exp(−2 k
x
d)
, (4)
and the amplification of the evanescent waves depends only on the condition on the dielectric constant (ǫ
2
= −ǫ
1
or ǫ
2
= −ǫ
3
). Such a system can be easily realized as several metals act as a good plasma in some range of optical
frequencies in that they can have a large negative real part of the dielectric constant with a comparatively small
imaginary part. Typically, we take our system to be a thin film of silver deposited on another medium such as glass,
GaAs, Silicon. We note that silver is highly dispers ive and by choosing the operating wavelength of light approprately,
the dielectric constant (ǫ
2
) of silver can be chosen to be either −ǫ
1
or −ǫ
3
. Note that µ = +1 everywhere for such a
system. In the ele ctrostatic limit, we can take the object plane and the image plane to be symmetric about the slab
at distance a d/2 from the edge of the slab. To arrive at a simple, though approximate, description of the asymmetric
lens, we will take the electrostatic limit in the remainder of this section. We will consider the effects of retardation in
Section IV.
The link between the amplification of the evanescent waves to the presence of a surface plasmon mode has already
been pointed out [2]. To obtain an insight into the process, let us examine the spatial field variation in our system.
First, let us consider the transmission (T ) and the reflection (R) from the slab as a sum of partial waves arising from
multiple scattering at the interfaces:
T = t
21
t
32
e
−k
x
d
+ t
21
r
32
r
12
t
32
e
−3k
x
d
+ t
21
r
32
r
12
r
32
r
12
t
32
e
−5k
x
d
+ ··· ,
=
t
21
t
32
e
−k
x
d
1 − r
32
r
12
e
−2k
x
d
, (5)
R = r
21
+ t
21
r
32
t
12
e
−2k
x
d
+ t
21
r
32
r
12
r
32
t
12
e
−4k
x
d
+ ··· ,
= r
21
+
t
21
r
32
t
12
e
−2k
x
d
1 − r
32
r
12
e
−2k
x
d
, (6)
where t
jk
= 2ǫ
j
/(ǫ
k
+ ǫ
j
) and r
jk
= (ǫ
j
− ǫ
k
)/(ǫ
k
+ ǫ
j
) ar e the par tial transmission and reflection Fr esnel coefficients
for P-polarized light (obtained by ma tching the tangential co mpone nts of E and H) acros s the interface between
Media-(j) and (k) (see figure 1 ). When the perfect-lens condition at any of the interfaces is satisfied, the partial
reflection coe fficient r
jk
as well as the partial transmissio n coefficie nt t
jk
for evanescence across the interface diverges.
It should be noted, however, that the S-matrix that relates the incoming and the outgo ing wave amplitudes in the
scattering process is analytic in the complex (momentum or energy) plane. Hence, the sum of the infinite series is
still valid due to the analyticity of the S-matrix, and the reflection and the trans mission from the slab a re well-defined
quantities. The above procedure is entirely equivalent to the usual time-honoured practice of decomposing the fields
into a complete set of basis states and matching the field amplitudes at the boundaries as follows. Let the fields in
regions – 1, 2, and 3 be given by
H
1
= e
−k
x
z
+ Re
k
x
z
, (7)
H
2
= Ae
−k
x
z
+ Be
k
x
z
, (8)
H
3
= T e
−k
x
z
, (9)
where R and T represent the reflection and transmission coe fficients o f the slab. Matching the tangential components
of H and E at the bounda ries, we o btain after some trivial algebra:
A(ǫ
2
= −ǫ
1
) =
ǫ
3
− ǫ
1
ǫ
3
+ ǫ
1
e
2k
x
d
, A(ǫ
2
= −ǫ
3
) = 0, (10)
B(ǫ
2
= −ǫ
1
) = 1, B(ǫ
2
= −ǫ
3
) =
2ǫ
3
ǫ
3
+ ǫ
1
, (11)
T (ǫ
2
= −ǫ
1
) =
2ǫ
3
ǫ
3
+ ǫ
1
e
k
x
d
, T (ǫ
2
= −ǫ
3
) =
2ǫ
3
ǫ
3
+ ǫ
1
e
k
x
d
, (12)
R(ǫ
2
= −ǫ
1
) =
ǫ
3
− ǫ
1
ǫ
3
+ ǫ
1
e
+2k
x
d
, R(ǫ
2
= −ǫ
3
) =
ǫ
3
− ǫ
1
ǫ
3
+ ǫ
1
. (13)
3
Note that there are two separate cas es, ǫ
2
= −ǫ
1
on the left side, and ǫ
2
= −ǫ
3
on the right side in the above equations.
[Insert figure 2 about here ]
First, we note that while the transmission is the same in both the cases, the spatial variation of the field is completely
different as shown in figure 2. Second, there is a non-zero reflection in both cases, and for ǫ
2
= −ǫ
1
the reflection
is also amplified. The first case of ǫ
2
= −ǫ
1
is exactly the condition for the excitation of a surface plasmon state
at the air-silver interface, where the field is seen to be large and decaying on either side of the interface. The other
case of ǫ
2
= −ǫ
3
corres ponds to the excitation of a surface plasmon state at the silver-GaAs interface. In this case
only the growing s olution within the silver slab is present. It is interesting to note that in the symmetric case of
ǫ
1
= ǫ
3
, again only this growing solution within the sla b is selec tively excited. The reflectivity is also zero for the
symmetric slab. To understand the role of the surface modes in the a mplification of the evanescent field, we note that
the condition ǫ
2
= −ǫ
1,3
is exactly that for a surface plasmon state to exist on an interface between the semi-infinite
negative and semi-infinite positive media. In a finite slab with two surfaces, however, the surface plasmons detune
from the resonant frequency, and the detuning is exactly of the right magnitude to ensure that they are excited by
incident fields to the correct degree for a focussed image.
The presence of a large reflectivity ha s se rious consequences for the use of a lens for near-field imaging applications
as it would disturb the object field. We call the first case of ǫ
2
= −ǫ
1
, where the reflection is also amplified, as the
unfavourable configuration, and the second case of ǫ
2
= −ǫ
3
as the favourable configuration of the asymmetric lens.
Obviously it would be most advantageous if the reflectivity were zero. Here we draw an analog y with a c onventional
lens which in the ideal has per fectly tra nsmitting surfaces, but in practice always pro duces some stray reflection from
the lens surface, but nevertheless still provides an acceptable degree of functionality.
III. EFFECTS OF ABSORPTION
Now we will consider the effect of absorption in the slab of the negative index medium on the imaging process. We
will work in the electrostatic limit or assume that the perfect lens conditions for the magnetic permeability (µ
2
= −µ
1
or µ
2
= −µ
3
) is also satisfied. Pendry [2] had shown that the e ventual resolution is limited by absorption in the lens.
This absorption can be included in the calculation by adding an imaginary pa rt to the die lectric cons tant. Pendry
had used the dispersion ǫ
2
(ω) = 5.7 −9.0
2
(¯hω)
−2
+ i0.4 for silver (¯hω in eV) and we will continue to use it here. Put
ǫ
2
= −ǫ
k
+ iǫ
′
2
, where k is either 1 or 3. The expression for the transmission becomes (assuming ǫ
′
2
≪ ǫ
k
),
T
p
=
−4ǫ
k
ǫ
3
exp(−k
x
d)
±iǫ
′
2
(ǫ
1
− ǫ
3
) − 2ǫ
k
(ǫ
1
+ ǫ
3
) exp(−2 k
x
d)
(14)
with the ± being chosen acco rding to k is either 1 or 3 respectively. The resolution in Pendry’s original case was
limited because for large k
x
the exponential term in the denominato r becomes smaller than the other term. Now, we
can see that there is definitely an advantage in choosing ǫ
k
to be the larger of ǫ
1
and ǫ
3
, in order to make the term
containing the exponential to dominate in the de nominator. Thus the asymmetr y can actually help us better the limit
on the re solution set by absorption. Fortunately, this also corres ponds to the favourable case of ǫ
2
= −ǫ
3
. Choosing
ǫ
k
to be the smaller of the two would, in contrast, cause degradation of the resolution. Hence, the sub-wavelength
resolution pr oposed earlier by us as the ratio of the optical wavelength to the linea r size of smallest reso lved feature
[8] works out to be
res = λ
0
/λ
min
= −
ln |ǫ
′
2
/2ǫ
3
|λ
0
4πd
, (15)
1 (a ssuming ǫ
3
≫ ǫ
′
2
andǫ
3
≫ ǫ
1
) in the favourable configuration.
[Insert figure 3 about here ]
Below we will consider the above effects of absorption on the image of two uniformly illuminated slits (intensity =1)
obtained in transmission by the asy mmetr ic slab of silver of thickness d =40nm. On the other side, we consider several
media of higher dielectric constant. We numerically obtain the ima ge at the image plane (co nsidered to be at d/2 in
the electro-static limit) and plot these in figures 3(a, b). In figure 3(a), we co nsider the case where the resolution is
enhanced due to the asymmetry in the lens. We take ǫ
2
= −ǫ
3
+ iǫ
′
2
by tuning to the appropriate frequency (lower,
according to the dispersion form), where ǫ
3
> 1 for all the media considered: water (ǫ
3
= 1.77), glass (ǫ
3
= 2.25),
zirconia (ǫ
3
= 4.6) and silicon (ǫ
3
= 14.9 at ¯hω
sp
= 2eV corres ponding to the surface plasmon being excited at the
silver-silicon interface). The leve l of absorption in silver is taken to be approximately the same at all these frequencies.
4
We consider the object to consist of two slits, 20nm wide and placed apart by a centre to centre distance of 80nm. We
find that the best contrast is obtained for the case o f silicon ra ther than the symmetric lens. In fact, the resolution
initially degrades with increasing ǫ
3
, but improves drastically for large values of ǫ
3
. In this case, the two slits are
not resolved fo r the c ase of g lass and zirconia. We obtain a limit of the spatial resolution of about 70nm (centre to
centre) for the case of silicon. In figure 3(b), we consider the images obtained for the case ǫ
2
= −ǫ
1
+ iǫ
′
2
, where
ǫ
1
= 1.0 for air on one side. This is the unfavourable configuration corresponding to the surfac e plasmon excitation
(¯hω
sp
= 3.48eV ) at the air-silver interface. We consider a larger object of two slits of width 30nm, the centres placed
apart by 120nm and find that the image resolution gets de graded with increasing ǫ
3
. For the case of Silicon (ǫ
3
= 16.4
at this frequency), the image is just resolved accor ding to the Rayleigh criterion
2
for resolution. However, we will not
be particularly interested in this case due to the amplified r eflected wave which will severely affect the imaging.
IV. EFFECTS OF RETARDATION
In the electrostatic(magnetostatic) limit of large k
x
∼ k
z
∼ q
z
, there is no effect of changing µ(ǫ) for the P (S)-
polarizatio n. The deviation from the electr ostatic limit caus e d by the non-zero frequency of the electromagnetic wave
would, however, not allow this dec oupling. We will now proceed to investigate the effects of retardation caused by
a finite frequency o f the light. It has already been noted [8] that a mismatch in the ǫ and µ from the perfect-lens
conditions would always limit the imag e resolution and also leads to large transmission r e sonances associated with
the excitation of coupled surface modes tha t could introduce artifacts into the image.
[Insert figure 4 about here ]
Let us first examine the nature of the sur face pla smons (for P-polar ized incident light) at a single interface between
a positive and a negative medium. The condition for the existence of a surface plasmon at the interface is [9]
k
(1)
z
ǫ
1
+
k
(2)
z
ǫ
2
= 0, (16)
which gives the dispe rsion relation:
k
x
=
ω
c
ǫ
2
(ǫ
2
− µ
2
)
ǫ
2
2
− 1
1
2
, (17)
where it is assumed ǫ
1
= 1 and µ
1
= 1 fo r the positive medium. Note that Eq. (16) can be s atisfied only for imaginary
k
(2)
z
, when ǫ
2
is negative. The dispersion is plotted in figure 4, where the causal plasma dispersion forms ǫ
2
= 1−ω
2
p
/ω
2
and µ
2
≃ 1 − ω
2
mp
/ω
2
are assumed for the material of the negative medium [7,6]. This is beca use a dispersionless
µ(ω) < 0 at all ω would not be physical [10]. We ca n see that the plasmon dispe rsion take different forms for ω
mp
> ω
p
(the upper curve) and ω
mp
< ω
p
(the lower curve). At large k
x
, the plasmon frequency tends to the electrostatic limit
of ω
p
/
√
2 from either above (ω
mp
> ω
p
) or (below ω
mp
< ω
p
). At small k
x
, the surface plasmon first appears at the
light-line (ω = ck
x
). We note that similiar results have been obtained by Ruppin very recently for the dispersion of
the surface plasmon modes in ne gative refractive media for the case ω
p
> ω
mp
[11].
[Insert figure 5 about here ]
Now, let us next consider the symmetric lossless slab (ǫ
1
= ǫ
3
, µ
1
= µ
3
= 1, ǫ
′
2
= 0) and examine the natur e of
these surface modes. As is well known, the two dege nerate surface plasmons at the two interfaces get coupled for a
thin slab and gives rise to coupled slab modes: a symmetric and an antisymmetric mode. The condition for these
resonances (for P-polar ized light) are [9]
tanh(k
(2)
z
d/2) = −ǫ
2
k
(1)
z
/k
(2)
z
, (18)
coth(k
(2)
z
d/2) = −ǫ
2
k
(1)
z
/k
(2)
z
, (19)
2
It should be noted that the Rayleigh criterion, in the strict sense, is applicable for the radiative modes. It states that two
separate sources are just resolved when the principal maximum of one image coincides with the first minimum in the diffraction
pattern of the other. This results in the ratio of the intensity at the mid-point between the sources to the intensity at the
maxima of ≃ 0.811. We will use this ratio as the equivalent measure for defining the resolution in our case
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