A summer project by
Nicholas A. Kuhta
Negative index of refraction media (NIM) shows many properties that
could lead to better near-field imaging applications. We used TM
polarized waves to calculate the resolution limit of the planar superlens,
and demonstrated that the excitation of H-field waves commonly occur at the
front and back interface of a planar superlens.
Project Description
The goal of this project was to study electro-magnetic waves as they traveled
through a planar lens that had a negative index of refraction (NIM). Electric
and magnetic fields were calculated by simultaneously solving the Maxwell equations.
Reflection and transmission coefficients were solved to establish a continuous field at
the boundaries of the material.
Fourier series methods were used to calculate the H-field as it passed from a source
through a planar NIM. While graphing the H-field we varied the width of the object, the
lens absorption, the separation between the object and the lens, and the lens thickness.
In this project we used the conventional superlens model, where the lens
thickness equals twice the distance from the object to the lens, which plays the role of a focal distance.
Note that the focal distance in this experiment is not the "conventional" focal distance because the
lens is not curved.
One-dimensional plots of the H-field’s magnitude were graphed at the image location, which is four
times the focal distance away from the object. By studying the field at the image
location one comes to the conclusion that it is feasible to resolve sub-wavelength features of an
object using a NIM system. A graph of the H-field at the image location is shown in Figure 1
Figure 1. H-field at the image
location
In addition to collecting data at the image location, we graphed the H-field propagating from the
source through the planar superlens. When the focal distance equaled the object size, graphs of the
H-field decayed from the source and then rose into two surface waves located at the boundaries of the lens.
This is shown below in Figure 2.
Figure 2. Graph of H-field along the
"focal line"
When the focal distance and lens thickness were lowered, the H-field decayed from the source and produced
one large surface wave at the back interface, and one relatively small surface wave at the front of the lens
as shown below in Figure 3
Figure 3. Graph of H-field when the
focal distance is smaller than the
object
The resolution of the planar
superlens depends only on the lens
thickness b, and the absorption
e". An approximation for the
resolution D is shown below
In order to study the appearance of
diffraction limit we simultaneously
increased the focal distance and
lens thickness. When the focal
distance was roughly 10 times larger
than the object chosen, the H-field
would decay from the source and
produce two maxima located in the
center of the planar lens and at the
image location. This is shown in
Figure 4.
Figure 4. H-field with focal length
10xObject size
Mathematica and Java were used to
solve and visually represent the
results of this project. In the
future similar techniques will be
used to study the imaging properties
of different lens configurations.
top
Playing with planar-imaging
Discover the physics of planar imaging by yourself. You need
Java installed on your machine to run the
applet. If you prefer not to install Java - please scroll down to see some
representative screenshots.
Setup the parameters in the input lines below, and click either "Calculate all"
or "Calculate 1D plots". Don't hurry - it may take up to several minutes for a
computer to simulate the full 2D imaging graph.
You can control:
- Size of the object [from 0.1
l to 2
l]
- The focal distance of the lens
(the lens thickness is twice its
focal distance) [from 0 to 15*object
size]
- Absorption in the NIM part [from
10-6 to 10-2]
(enter the exponent of absorption)
- The blue one dimensional plot
represents a cross section of the
H-Field in the yz plane.
- The red one dimensional plot
represents a cross section of the
H-Field in the xy plane located at
the image.
- The dashed yellow lines on the
density plot represent the boundary
of the planar lens.
top
Some representative
screenshots of
the applet:
Super-imaging regime. The intensity
maxima coincide with lens
boundaries.
Transition to the
diffraction-limited regime. The
intensity has its maxima at the lens
boundaries, accompanied by the
maxima at the "focal points" inside
and behind the lens.
Diffraction-limited imaging.
Intensity maxima coincide with focal
points.
top
