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Physical principles, mathematical treatment and
realization of a new near field microscope in the
11
13
THz region (10 -10 Hz)
D.Coniglio, A.Doria ENEA, FIS-ACC via Enrico Fermi 45, 00044 Frascati (Rome) ITALY
TeraHertz
Gap
Near Field Microscopy:
Propagating beam
reemitted by the
nanocollettor
Propagating
wave
Incident
light beam
Wavelength l
3000 mmmm 15 mm
Frequency n
1011-1013 Hz
Energy E
0.4meV-80meV
Evanescent beam
generated the object
surface
Nanocollettor
Object
surface
Evanescent beam
generated by the
nanocollettor
One
wavelength
Near field microscopy is based on
the detection of evanescent fields.
They are confined on the object
surface and related with its
nanostructure. For their non
propagating nature
nature, it’s not
possible to detect them far away
from the sample (i.e. far field
region). It’s necessary to put very
near to the object’s surface (i.e.near
field region) a small scattering
element, the nanocollettor wich is
able to capture these fields and
convert them into propagating ones.
The field detected preserves the
informations about the object
subwavelength structure.
physical principles
The detection of evanescent fields
makes possible beating the Abbe
diffraction limit, which states that in
optics it’s impossible to get a
subwavelength
subwavelength resolution. This
detection is based on the optical
frustration principle: onto a prism the
total internal reflection can be avoided if
a second prism is brought very near to
the first one.This phenomenon is called
optical tunnel effect: on the surface of
the first prism exists an evanescent field;
if a suitable dielectric material is
immersed in it, this one will be
converted into a propagating field in
order to respect the continuity conditions
at the interface.
d << l
A general structure of such non-propagating fields is this one
U(x  y  x  t)
Subwavelength
aperture: Bethe’s
theory
(
) e
 j kxxkyy  z jt
A(x  y  z) e
Amplitude
field
Long wavelength
Near Field Microscopy
Visible-Infrared near
field microscopy
e
Evanescent
term
Propagation term
Time dependence
Equivalent principles
Waveguide with a
localised source
??
l
d ??
??
In 1942 the physicist H.A.Bethe solved with a mathematical theory, the
problem of diffraction from a hole which dimensions are negligible
respect to the incident radiation wavelength.
 n  (E  H
He wanted to estimate the effects of a small hole made on a common
conductive wall of two coupled waveguides.
S
He supposed that in order to preserve boundary conditions on the hole
and on the conductive plane, fictitious electric and magnetic dipoles
have to exist instead of the hole. From this point of view the field
diffused by the aperture is equivalent to the field irradiated by these
dipoles; for this reason they are called equivalent dipoles.
dipoles
Equivalent electric and
magnetic dipoles


Hl  J m da
hole
If we suppose that hole’s dimensions are negligible respect to the
wavelenght, we can expand in Taylor series the field H; unlikely from what
happens when the field E is expanded, in this case the zero order term is
responsible of a magnetic coupling and the first order term of an electric
one. So both the dipole’s moments have the magnetic and electric
interactions exchanged. Using the expression for Jm , we finally can write the
expressions for the equivalent dipoles.
Hole radius
Trasversal field H
rr 2 3 rr
P  r0 e0 (n0 E)n0
3
rr 4 3
M  r0 Ht
3
Stored energy and electromagnetic field produced by
both the dipole
Z (mm)
Fields
r
q
n
Z
f
M
X
Electromagnetic fields in tapered
metallic waveguide
d
Hb
dz
d
Hf
dz
(
1 d
1 d
iK1(z) Hb 
 a (z) Hb 
 K1 (z)  Hb  Hf
a(z)d z
2 K1(z) d z
(
)
)
1 d
1 d
 a (z) Hf 
 K1(z)  Hf  Hb
iK1(z) Hf 
a(z)d z
2K1 (z)d z
Propagation term
Decrease (increase) of magnetic field in
Mutual reflective coupling of forward and
a larger (narrower) structure
backward waves
Energy
P
These expressions have been derived in the near field limit (r << l). Let’s
notice that they are exactly the same expressions known for the fields due to
static electric and magnetic dipoles.
So if we calculate the Poynting vector of these fields we obtain exactly zero
because they don’t carry out energy;
energy they only have a stored energy (E2,
B2).
We calculate the electromagnetic fields in a tapered waveguides by
solving a system of two coupled differential equations which describe
the local reflection of the waveguide mode as the diameter changes
gradually. The model can be applied both in the cut-off and in the
propagating region.
V
Y
1
E(J,f,r) [n(J,f)(3Pn(J,f))P] 3
4e0r
Electric and magnetic fields produced by
electric dipole; the fields produced by
magnetic dipole are obtained substituting E,
B, P with B, -E, M.
 El  H )da   J  El d x
 3
2 Z l
 3
Al  
J  El d x

V
c
Y (mm)
cm0e0
B(J,f , r)  ik0 (n(J,f )  P) 2
4r
l


Dipoles’ electromagnetic
field: stored energy
If on the guide’s walls, between S+ and S-, there’s an aperture, (longitudinal
coupling), we have to add this term in the right side of the expression for A l

Distance from the waveguide horizontal plane
0.22mm
X (mm)
Z (mm)
Wavelenght
2.6mm
Hole radius
0.13mm (l\20)
Waveguide dimensions
1mm-4mm
Waveguide fundamental mode
TE 10
Results
TeraHertz configuration:the metallic tip is used in the cut-off region
in order to produce an evanescent filed (plot on the left).
Millimetric configuration: the collector works in the propagating
region; in these calculations dielectric losses are not included (plot on
the right).
forward
backward field
100
10
1
0.1
0.01
1103
1104
1105
1106
1107
1108
Wavelenght
700 m m
Wavelenght
2.6 mm
1
Tip radius
70 m m (l\
(l\10)
10)0.5
Collector
radius
1.05 mm
Taper
full angle
10 degree
Tip full angle
20 degree
0
Tip length
5.5 mm
640 540 440 340 240 140 40
mm
0
2
4
6
8
Coupling coefficient between near
field and waveguide modes versus
modes index.
Collector length
4.5 cm
References
H.A.Bethe, Theory of diffraction by small holes Phy.
Rev. Vol. 66, pp 163-182, 1944
Collin, Foundation for microwave engennering.
S.A.Shelkunoff, Field equivalence Theorems,
Comm. On Pure and Appl. Math, vol.4 pp. 43-59,
1951
Electromagnetic Field in the cutoff regime of tapered
metallic waveguides. B.Knoll,F.Keilmann. Opt.
Comm. 162