|
|
Surface physics and holography
Prof. Alessandro Marvin, Associate Professor
You find below a list of arguments tackled in almost fourty years of
research, starting from the times when the Surface Physics was a novelty.
More recently my interest was mainly centered on a mathematical reformulation
of the "Theory of Optics" starting from the Geometrical Optics
which I derive through the angular eikonal as Landau did in his textbook
" The Classical Theory of Fields ",
then introducing the diffraction, visualized here as a bare consequence
of the wave character,
focusing - with many reported examples - on the Abbe barrier
and the related concept of
the Numerical Aperture - both too many times done for granted
or treated with not sufficient clarity in treating Lens Optics -, then
ending with the main purpose of the book which is and
remains the subwavelength resolution
in Near-Field Optics.
With this I intended to put a link between the
Far-Field Optics and the Near-Field Optics both treated in the same footing.
Surprisingly the simple Kirchhoff approximation I use throughout,
is capable to explain most of the NSOM results and correctly leads to a
resolution which is close to the experimental value. This work, which
claims something very different of what stated in the litterature
where many experimentalists force, on the contrary, to present the
NSOM results as a " Magic Mistery ",
far from being explained theoretically, remains
- as most of my best papers - unpubblished. With no intention to fight
with the editors I refer it to
the
Research Section of my Home Page for whoever is interested in.
A. M. Marvin
Trieste, Sept. 25, 2010
|
My starting and pioneering works on surfaces.
These topics all cover the scattering from surfaces,
a wide branch of Surface Physics.
Most of the theoretical work has been done using light
as the external probe, but optical emission using electrons and
diffraction of light atom over a crystal surface
are considered also.
For light
one distinguish two cases. The first is the scattering from
1 Dim gratings or 2 Dim rough surfaces [2,3] both in the small
roughness limit: the surface is
"statically"
corrugated and the scattering is elastic (diffraction).
The second arises
from the interaction with bulk and surface phonons: the surface is
"dynamically"
corrugated and the scattering is anelastic (Brillouin scattering).
Here the surface appears as a ripple on a flat surface [1]
or superimposed on an already corrugated grating [5].
From a theoretical point of view, and unless for a change of nomenclature,
the two cases look similar, thus treated with the same machinery
(extinction theorem).
This holds for surface modes i.e. resonances one is mostly interested in.
For diffraction one deals with Surface Polaritons/Plasmons (SP)
excited by light via the surface corrugation,
then decaying into the radiative channels.
This is a second order effect in the roughness, thus faint in the
small roughness limit, but nevertheless crucial for the observed anomalies
in the diffracted channels. For a grating, and plotting the deep
in the reflected spectral intensity
in terms of the incident angle, one gets informations on the
dispersion relation of SP mode on a flat slab.
The same coupling is present between the two SP modes at the zone border.
Here a frequency gap opens, and whose width is of the first order
in the roughness.
In Brillouin scattering the incoming light looses much of its energy
in the Raylegh wave (RW). Leaving out the discussion of my pioneering work
done around seventies in order to interpret the spectra in terms of the
ripple and elasto optic contributions, the most inportant results are
gotten again on a grating instead of a flat slab.
The SP and RW mix together forming a RW ``replica''
in the Brillouin spectra
[Marvin, Nizzoli, Phys Rev B 45 12160 (1992)]]. In addition
the grating periodicity couples the surface mode to the bulk continuum
opening a frequency gap in the RW at the zone border [5].
The same resonances in the cross section are observed using
electrons as a tool.
The electrons can be used in two ways depending of their energy,
and both measure the SP at large momenta (no retardation).
The first is the Electron Energy Loss Spectroscopy (EELS).
Low energetic electrons of a few eV do not penetrate into the crystal and are
reflected from the surface. Much of the energy loss is on the SP, whose
dispersion relation can thus be measured.
The experiment was done by the Genoa group, but in my opinion their results
remain still unexplained
[Marvin and Toigo, Europhys. Lett.,14 (5) 445 (1991)] .
The second method (SPREE) is to use high energetic electrons (which
do trespass the slab) on a rough surface. What is measured is the
optical emission which, in addition to the Transition Radiation, shows a
peak in the intensity at the Surface Plasmon frequency in the large K limit.
This is due to the excited SP mode which decay into the radiative channels
via the roughness. The ``Kretschmann splitting''[Kretschmann, et al.
Phys Rev Letters 42 1312 (1979)]
expected to occur from the degeneracy of the SP modes
at large momenta, has been never observed till now. This is
probably due to the surface autocorrelation function (ACF) which is
not gaussian as supposed [3].
Full agreement with experiment is gotten instead for SPREE on a grating.
Here the electron excited SP mode decays via grating into a
radiative channel increasing its intensity.
Plot of the emission peaks from constant frequency angular scans,
shows at the zone border a momentum-gap (k-gap)
instead of the usual frequency-gap which occurs with a light beam.
This result is perfectly reproduced by the theory [4].
Neutral atoms
(mainly He4) are the best candidates to study the crystal
surfaces since, in contrast to the electrons, the atoms do not penetrate
bejond the first layer.
However a great precision is required to the apparatus for determining
the surface structure. For this, the experiments using He4 and Ne
(quantum limit) have become possible only around seventies by the Genoa group.
In addition to the surface structure, the cross section gives a lot of
information on the gas-surface potential. This is constructed through the
energy position of bound states present in the attractive part of
the potential. A fine scale analysis of the spectra contains this
information, since they couple to the diffracted channels via the lattice
periodicity of the crystal surface. The pioneering work was done by Levi and
I have contributed only occasionally for what concers the temperature
dependence.
The most relevant pubblications.
- " Brillouin Scattering from Surface Phonons in Aluminium
Coated Semiconductors "
- V. Bortolani, A. M. Marvin, and J. R. Sandercock,
Phys. Rev. Lett. 43, 224 (1979)
- " Optical Properties of Rough Surfaces;
General Theory and Small Roughness Limit "
- F. Toigo, A. M. Marvin, V. Celli, and N. R. Hill,
Phys. Rev. B15, 5618 (1977)
- ("collected paper" for the Optical Society of America,
Ed. J. Bennett, 1992)
- " Resonant Light Scattering from a Randomly Rough
Surface "
- G. C. Brown, V. Celli, M. Haller, A. A. Maradudin, and A. M. Marvin
- Phys. Rev. B31, 4993 (1985)
- ("collected paper" for the Optical Society of America,
Ed. J. Bennett, 1992)
- " K-Gaps for Surface Polaritons on Gratings:
Excitation by Fast Electrons "
- P. Tran, V. Celli, and A. M. Marvin,
Phys. Rev. B42, 1 (1990)
- " Theory of Surface Acoustic Phonons Normal Modes
and Light Scattering Cross-Section
- in a Periodically Corrugated Surface "
- L. Giovannini, F. Nizzoli, and A. M. Marvin,
Phys. Rev. Letters 69, 1572 (1992)
Relatively more recent works.
Inelastic resonances and sticking in atom
surface scattering. (Celli, Marvin)
In the extreme quantum limit of diffraction, light atoms
(usually Helium 4) see the surface as a hard corrugated wall
with an attractive well in front. The corrugation gives rise to
diffraction (change in the parallel momentum K), while the attractive part
allows the interaction with the bound states of the well.
The change in K implies a
change in the perpendicular mumentum kz within the well,
thus, let's say, on the "perpendicular energy" Ez.
For -D < Ez < 0 where D is the depth of the well and if
Ez is equal to energy of a bound state, on has a resonance.
The atom is trapped for a long time before emerging back
from the surface. In this conditions the distorted wave Born approximation
(DWBA) fails, and the temperature dependence
(Debye Waller factor) is hardly acconted for going bejond the DWBA.
A recipe normaly used is to start writing the interative two-potentials
scattering equation for the T matrix as would be in the absence of phonons,
in term of the reflection coefficient R of the well
and the S matrix for the wall.
Next step is to interpret each elastic matrix element S for the collision
as the phonon averaged counterpart and multiply it
by the appropriate ``square root of the Debye Waller factor'' obtained
in the DWBA.
Surprizingly this prescription works in many cases. Even if quite
reasonable from an intuitive point of view, it is difficult to justify it
theoretically. In addition using this recipe,
it is not clear which phonons are involved
in the trapping and how they are contributing to the final scattering.
This point has been clarified.
We have show [1] that exactly the same but more general equations
are obtained introducing a width in the bound state propagator.
This implies:
first, any photon absorbed in the bound state
can be reemited coherently only within a time interval of the order the
inverse width. In other words dephasing is accounted for automatically.
Second, multiple reflections (infinite) in the well are summed up.
Third, no phonon exchange is allowed in entering and exiting from a
bound state, and this is the only restrictions we made for the solution.
The theory violates the unitarity thus, as we believe,
includes the sticking.
Atomic holography: the polarized light
improvement. (Bortolani, Celli, Marvin)
All the holographic techniques originate from the Gabor's original idea to
build a microscope without lenses and are based on the interference
between optical paths.
In the atomic holography ("fluorescence holography")
the radiation (reference wave) is generated by an
excited atom (emitter) and then scattered by neighbouring atoms.
Recording this radiation on a photofilm (detector), one gets the
so called ("optical") hologram which contains
information on the positions of the atoms in the structure.
In the course of time the photofilm has been substituted with a digital
detector. Second, the ``decoding wave'' previously used to
illuminate the photographic film and reconstructing the image,
has been supressed in favour of a numerical procedure.
This is essentially a Fourier Transform (FT)
of the hologram data over an energy shell in k-space.
Third, the emitter is going to play the role of a detector,
while the reference wave is furnished by a synchrotron radiation source.
For simplicity numerical simulations are performed using a scalar theory,
but the light polarization does play in fact a role in the image
reconstruction. We have shown how the best imaging is obtained with a beam
polarized in the direction perpendicular to the line joining that atom
to the emitter. The worst is gotten rotating it arond the beam direction
by 90 degrees [2]. The unpolarized beam (Thomson scattering) gives
an average of these two images.
We suggest how to get better images using a linear polarized light beam
instead of an unpolarized one. The precedure is rather simple.
First perform three measurements (instead of a single one)
on the sample, rotating the polarization by 90 and 45 around the beam
direction for the last two (the polarization in the first measurement
is arbitrary).
Second, construct the ``optimal'' hologram for the best observation
at a given atom at ra as a linear combination
of the three holograms.
Third, and as for unpolarized beam, use the FT technique but on the optimal
hologram. In practice one interpolated between ra atomic
positions (unknown) thus the optimal hologram becomes in addition to a
k-dependent also a position r-dependent quantity.
To see how the method works, are here reported two figures. The first
(click
here)
is the polarized light image
(the three arrows start from the emitter position on the left); the second
(click
here)
is the reconstruction gotten with an unpolarized beam.
In figure 1 all the three atoms are imaged and well centered; in figure 2
only one atom appears in the reconstruction, while the other two are missing.
The advantage of using a polarized beam clearly appears
at low energy as that used here, but the same remains true
at any energy.
From Surface Physics
to Differential Geometry:
application to antenna radiowaves and others.
(Celli, Marvin)
The physical aspect of the problem is the light scattering from rough
surfaces we start with. The mathematical one is the theory of surfaces
we are led to. The connection between the two is the surface impedance Z
expressed in terms of the extrinsic curvature tensor of the surface [3].
As known, the surface impedance postulates the existence of a
local relation between the parallel components of the electric
and magnetic fields on the surface.
The advantage of having such a relation is easily seen since
the number of equations is reduced by a factor of two. On one side
the analytical solution (small roughness) becomes easier;
on the other, a numerical simulation, othervise limited
to a one dimensional (1D) gratings,
can be applied to stochastic 2D corrugated surface too.
Unfortunatelly this relation is not local since the exact boundary
condition (extinction theorem) involves the fields as well as
their derivatives, the last being present even for a perfectly flat geometry.
However, locality is correct to first order in the
penetration depth d, thus applies with a good approximation
to metals both at infrared and optical frequencies.
Corrections (local and not local)
have been evaluated up to third order in d and found again in terms of the
extrinsic curvature tensor and its (first and second order)
absolute differentiation [4].
These results are the major contribution of our work,
and the way we get them is interesting by itself.
The philosophy is to carry out all the operations in the 3D space,
then projecting the results on the 2D surface. Vectors and tensors in 2D
are thought as the projected part of vectors and tensors in the surrounding
3D world. The differentiation of scalars and tensors
are performed in terms of the projector operator. This replaces the covariant
and contravariant derivatives but, using Cartesian axes, Christoffel's
symbols are avoided from the start. The method offers the possibility to find
important relations such as the exchange of differentiation of a vector.
It is remarkable to note that in this operation the Riemann tensor
appears directly into the result and has not to be compared with the
Christoffel's symbols, as one is forced to do in the Theory of Surfaces.
The same holds for Mainardi-Codazzi equations involving the derivative
of the curvature tensor.
This is why we call this method symbolic: it allows to
find important relations without entering into a detailed calculation.
The advantage of the present method lies also in
the immediate visualizability of the quantities that enter into the
Theory of Surfaces. Thus, looking from the outside world,
the concept of a parallel transport becomes obvious,
and the geodesic appear the natural ``straight'' line.
In terms of the projector the equations become very easy to write down,
and the geometric meaning of absolute differentiation (covariant and
contravariant) better understood.
The method does not prevent the distinction between ``intrinsic'' and
``extrinsic'' properties, but the two quantities are generally treated
in the same footing in the embedding space. The Theorema Egregium and
the Gauss-Bonnet theorem can be equally derived using the
symbolic method.
If interested, more details you find visiting the item
" Differential Geometry in Surface Optics: the
symbolic method ",
placed on the top of the page you open with this link.
- " Surface resonances and sticking
in the Hutchison model "
- V. Celli and A. M. Marvin,
J. Phys.:Cond. Matter 14, 6147 (2002)
- " Multiple-energy x-ray holography:
Polarization effect "
- V. Bortolani, V. Celli, and A. M. Marvin,
Phys. Rev. B 67, 24102-7 (2003)
- " The general relation between surface impedance
and surface curvature "
- T. T. Ong, V. Celli, and A. M. Marvin,
J. Opt. Soc. Am. A11, 759 (1994)
- " Relation between the surface impedance and
the extintion theorem on a rough surface "
- A. M. Marvin and V. Celli, Phys. Rev. B50, 14546 (1994)
e-mail: marvin@ts.infn.it
(Personal Pages: Research Section)
.
|
![[ Department of Theoretical Physics ]](/images/Dft.gif)