[ Department of Theoretical Physics ]





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.

    1. " Brillouin Scattering from Surface Phonons in Aluminium Coated Semiconductors "
      V. Bortolani, A. M. Marvin, and J. R. Sandercock, Phys. Rev. Lett. 43, 224 (1979)

    2. " 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)

    3. " 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)

    4. K-Gaps for Surface Polaritons on Gratings: Excitation by Fast Electrons "
      P. Tran, V. Celli, and A. M. Marvin, Phys. Rev. B42, 1 (1990)

    5. " 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.
    1. " Surface resonances and sticking in the Hutchison model "
      V. Celli and A. M. Marvin, J. Phys.:Cond. Matter 14, 6147 (2002)

    2. " Multiple-energy x-ray holography: Polarization effect "
      V. Bortolani, V. Celli, and A. M. Marvin, Phys. Rev. B 67, 24102-7 (2003)

    3. " 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)

    4. " 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) .


    Author: A. Marvin
    Updated : 03 Sep 2010
    URL: http://www-dft.ts.infn.it/DFT/activities/marvin.html