
Surface physics and holographyProf. Alessandro Marvin, Associate Professor
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 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].
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 momentumgap (kgap) instead of the usual frequencygap which occurs with a light beam. This result is perfectly reproduced by the theory [4].
The most relevant pubblications.
Relatively more recent works.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 k_{z} within the well, thus, let's say, on the "perpendicular energy" E_{z}. For D < E_{z} < 0 where D is the depth of the well and if E_{z} 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 twopotentials 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.
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 kspace. 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 r_{a} 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 r_{a} atomic positions (unknown) thus the optimal hologram becomes in addition to a kdependent also a position rdependent 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.
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 MainardiCodazzi 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 GaussBonnet theorem can be equally derived using the symbolic method. If interested, more details you find visiting the item placed on the top of the page you open with this link.
 

