Surface Profile Function Research Articles

Electrostatic surface shape resonances of a finite number of ridges

The resonance properties of localized electrostatic surface modes associated with a finite number of ridges on an otherwise planar surface are investigated. Numerical solutions of the homogeneous integral equations that describe the electromagnetic fields in the vicinity of the ridges are used to obtain the dispersion relation of surface plasmons. The frequencies of the electrostatic surface shape resonances are calculated for ridges with Gaussian, Lorentzian, sinusoidal, exponential, and triangular profiles. We show the existence of splittings of the plasmon frequencies, which depends on the surface profile function and on the distance between the ridges. Considering the ridge with a sinusoidal profile, we obtain the limit on the number of ridges which generates a frequency splitting of the electrostatic surface shape resonances, whose frequency values converge to those of the dispersion relation of surface plasmons on one-dimensional sinusoidal grating.

Enhanced backscattering of surface polaritons on a randomly rough surface

We present a theory of the enhanced backscattering of surface polaritons on a two-dimensional randomly rough surface. We solve the scattering problem by the method of reduced Rayleigh equations using unitary and reciprocal expansions in powers of the surface profile function. We calculate the angular dependence of the intensity of the surface polaritons into which the incident surface polariton has been scattered, and find that it has a peak in the backscattering direction. The effect is expected to be most easily observed by the methods of near-field optical microscopy in the infrared range.

X-ray scattering from a randomly rough surface

On the basis of the method of reduced Rayleigh equations we present a simple and reciprocal theory of the coherent and incoherent scattering of x-rays from one- and two-dimensional randomly rough surfaces, that appears to be free from the limitations of earlier theories of such scattering based on the Born and distorted-wave Born approximations. In our approach, the reduced Rayleigh equation for the scattering amplitude(s) is solved perturbatively, with the small parameter of the theory η(ω) = 1 - ϵ(ω), where ϵ(ω) is the dielectric function of the scattering medium. The magnitude of η(ω) for x-rays is in the range from 10−6 to 10−3, depending on the wavelength of the x-rays. The contributions to the mean differential reflection coefficient from the coherent and incoherent components of the scattered x-rays are calculated through terms of second order in η(ω). The resulting expressions are valid to all orders in the surface profile function. The results for the incoherent scattering display a Yoneda peak when the scattering angle equals the critical angle for total internal reflection from the vacuum-scattering medium interface for a fixed angle of incidence, and when the angle of incidence equals the critical angle for total internal reflection for a fixed scattering angle. The approach used here may also be useful in theoretical studies of the scattering of electromagnetic waves from randomly rough dielectric-dielectric interfaces, when the difference between the dielectric constants on the two sides of the interface is small.

Perturbation theory results for the diffuse scattering of light from two-dimensional randomly rough metal surfaces

Abstract Using the unitarity and reciprocity preserving formulation of Brown et al a perturbation treatment, correct to fourth order in the surface profile function, is given for the scattering of electromagnetic waves from a weakly rough, two-dimensional, random metal surface. In this formulation the boundary conditions on the electromagnetic fields are satisfied using the extinction theorem in conjunction with the Rayleigh hypothesis and the vector equivalent of the Kirchhoff integral. The theory is applied to, and results are presented for, several different types of rough surfaces which are characterized by power spectra that are extensions to two-dimensional random surfaces of the power spectrum of some one-dimensional random surfaces recently fabricated by West and O'Donnell. These surfaces, which can be realized experimentally, favor coherent, interferent, multiple scattering of electromagnetic waves via surface plasmon polaritons in intermediate states, and clearly exhibit enhanced backscattering ...

Surface plasmon polariton mechanism for enhanced backscattering of light from one-dimensional randomly rough metal surfaces

Recent experimental results of West and O’Donnell [ J. Opt. Soc. Am. A12, 390 ( 1995)] for the enhanced backscattering of p-polarized light from weakly rough, one-dimensional, random gold surfaces are compared with the predictions of two perturbative calculations of such scattering. The experimental surfaces were fabricated to possess power spectra that are nonzero in only a narrow range of wave numbers about the wave number of the surface plasmon polariton supported by them at the frequency of the incident light. As a consequence, enhanced backscattering that is due to the coherent interference of time-reversed scattering sequences involving counterpropagating surface plasmon polaritons is possible for only a limited range of values for the angles of incidence and scattering. The perturbative calculations used in the comparisons with experiment are the infinite-order calculation of McGurn et al. [ Phys. Rev. B31, 4866 ( 1985)] in the small-roughness approximation and the small-amplitude perturbation theory of Maradudin and Me′ndez [ Appl. Opt.32, 3335 ( 1993)] that is exact to fourth order in the surface-profile function. In both calculations the origin of enhanced backscattering is the coherent interference of multiply scattered surface plasmon polaritons with their time-reversed partners. The good quantitative and qualitative agreement between the theoretical and experimental results, with no fitting parameters, supports the conclusion of West and O’Donnell that their data demonstrate the existence of enhanced backscattering caused by the excitation of surface plasmon polaritons on a weakly rough random metal surface.

Scattering of a scalar wave from a two-dimensional randomly rough Neumann surface

Abstract We present a reciprocity and unitarity preserving formulation of the scattering of a scalar plane wave from a two-dimensional, randomly rough surface on which the Neumann boundary condition is satisfied. The theory is formulated on the basis of the Rayleigh hypothesis in terms of a single-particle Green's function G(q‖|k‖) for the surface electromagnetic waves that exist at the surface due to its roughness, where k‖ and q‖ are the projections on the mean scattering plane of the wave vectors of the incident and scattered waves, respectively. The specular scattering is expressed in terms of the average of this Green's function over the ensemble of realizations of the surface profile function (G(q‖|k‖)). The Dyson equation satisfied by (G(q‖|k‖)) is presented, and the properties of the solution are discussed, with particular attention to the proper self-energy in terms of which the averaged Green's function is expressed. The diffuse scattering is expressed in terms of the ensemble average of a two-p...

Anderson localization of surface electromagnetic waves on rough, perfectly conducting surfaces that are periodic on average

Based on the integral equation satisfied by the Green's function G( r o , r) for electromagnetic waves on a one-dimensional, randomly rough, perfectly conducting surface that is periodic on average, a numerical simulation approach is developed that yields the function 〈| G( r 0, r)| 2〉 − |〈 G( r 0, r)〉| 2, where the angle brackets den the ensemble of realizations of the surface profile function. The frequency dependence of the localization length, obtained from the exponential decay of this function, is then calculated for frequencies about the edges of the surface electromagnetic wave bands of the underlying periodic system. In the gap, the localization length is very small and fairly independent of the frequency. However, when the frequency enters in the allowed bands, the localization length becomes larger; this increase is found to be steeper the weaker the disorder is.

The impedance boundary condition at a two-dimensional rough metal surface

We consider the system consisting of vacuum in the region x 3 > ζ( x ‖) and a metal characterized by an isotropic, frequency-dependent, dielectric function ϵ(ω) in the region x 3 < ζ( x ‖) . The surface profile function ζ( x ‖) is a single-valued function of the vector x ‖ = (x 1, x 2, 0) that is differentiable as many times as is necessary. By the application of the vectorial form of Green's second integral identity to the region filled by the metal a pair of coupled integral relations between the two independent components of each of the vectors J E( x ‖|ω ) = n × E(x|ω)| x 3=ζ(x ‖) and J H(x ‖|ω) = n × H(x|ω) | x 3=ζ(x ‖) is obtained, where E(x|ω) and H(x|ω) are the frequency Fourier components of the macroscopic electric and magnetic fields in the system, while n = ( −∂ζ(x ‖) ∂x 1 , −∂ζ(x ‖) ∂x 2 , 1) . A nonlocal relation between J E(x ‖|ω) 1,2 and J H ( x ‖| ω) 1,2 is implied by these relations, and a local relation of the form J E ( x ‖| ω) i = K ij (0)( x ‖| ω) J H ( x ‖| ω) j ( i, j = 1,2) is extracted from it. The two coupled integral equations satisfied by the elements of the 2 × 2 matrix K ij (0)( x ‖| ω) are solved analytically to yield K ij (0)( x ‖| ω) as an expansion in powers of the skin depth of the metal at the frequency ω of the electromagnetic field. The first two terms in this expansion are presented. The way in which nonlocal corrections to the relations between J E ( x ‖| ω) 1,2 and J H ( x ‖| ω) 1,2 can be obtained is also outlined.

Enhanced backscattering of light from weakly rough, random metal surfaces

The enhanced backscattering of p-polarized light from a small-rms-height, small-rms-slope, onedimensional, random metal surface was first predicted on the basis of an infinite-order perturbation calculation that used the techniques of many-body theory and the concepts of weak localization theory. We present an elementary calculation of the contribution to the mean differential reflection coefficient from the incoherent component of the scattered light that is based on the first two nonzero terms in the expansion of this function in powers of the surface-profile function. We show that this approach not only accounts for enhanced backscattering but also gives the correct order of magnitude of the effect predicted by infinite-order perturbation theory.

The impedance boundary condition for a periodically corrugated metal surface

We present a method based on Green's second integral theorem for obtaining a local impedance boundary condition for a p- or s-polarized electromagnetic field at the periodically corrugated interface between vacuum and a metal. The surface is defined by the equation χ 3 = ζ ( χ 1), where the surface profile function ζ (χ 1) is a single-valued, periodic function of χ 1, and the plane of incidence is the χ 1χ 3-plane. The results are used in the determination of the dispersion curves for surface plasmon polaritons propagating normal to the grooves and ridges of the surface.

Angular correlation functions of amplitudes scattered from a one-dimensional, perfectly conducting rough surface

We study theoretically the angular dependence of the correlation functions of the scattering amplitudes occurring in the interaction of a beam of polarized light with a one-dimensional, perfectly conducting rough surface. For an ensemble of surface-profile functions that are realizations of a stationary stochastic process, a necessary condition for the exact scattering amplitudes to be correlated is established: the projection on the mean surface of the difference between the incident wave vectors must equal that of the scattered wave vectors. The exact expressions for the amplitudes of the scattered plane waves constituting the far field are derived from Green’s second integral theorem. By numerical simulations, the dependence of these amplitude correlation functions on the angle of incidence and on the incident and the scattered polarization states is computed for Gaussian surfaces producing enhanced backscattering. Results are presented for the complex correlation functions of p- and s-polarized scattering amplitudes. However, it is argued that, for an incident field polarized at +45°, the single- and the multiple-scattering contributions to the amplitude correlation functions are clearly separated if the scattered field is resolved into −45° and +45° polarized plane waves. In this case the real part of the correlation function of the −45° scattering amplitudes displays two peaks of large angular width. The maxima occur for angles of incidence at which the correlation between the +45° polarized scattering amplitudes has sharp peaks of angular width approximately equal to λ/a, where the surface correlation length a may be interpreted as an estimate for the mean distance between successive scattering points on the surface. One of these maxima occurs when the correlated amplitudes are identical, and the other occurs when the role of the incoming and the outgoing directions is interchanged between the two scattering amplitudes, which are then related by the reciprocity condition. The coherent addition of the amplitudes arising from multiple-scattering processes and those of their time-reversed partners in directions close to the retroreflection direction is demonstrated in the calculations of the correlation functions.

Propagation of shear horizontal surface acoustic waves parallel to the grooves of a random grating

An elastic medium that occupies the region x3≳ζ(x1), where the surface-profile function ζ(x1) is assumed to be a stationary, stochastic process, is studied. The displacement field u(x;t) in this region is assumed to satisfy stress-free boundary conditions on the surface x3=ζ(x1). First the boundary conditions satisfied by the mean displacement field 〈u(x;t)〉 on the plane x3=0 are obtained, where the angle brackets 〈 〉 denote an average over the ensemble of realizations of ζ(x1). The results are then used to obtain the dispersion relation for surface acoustic waves of shear horizontal polarization propagating in the x2 direction, i.e., parallel to the grooves of the random grating defined by the surface-profile function ζ(x1). It is found that the surface acoustic waves of this polarization, which cannot exist on a planar, stress-free surface of an elastic medium, are trapped by the random roughness, but are bound to the surface for only a finite range of values of the wave number k2 characterizing their propagation in the x2 direction. The attenuation of these roughness-trapped surface acoustic waves is also determined.

Enhanced backscattering in a magnetic field

Abstract By means of numerical simulations the authors study the scattering of a beam of p-polarized light from a small RMS slope one-dimensional random surface on a semi-infinite metal or n-type semiconductor to which a constant magnetic field is applied. The surface is defined by the equation x 3=ξ(x 1), where the surface profile function ξ(x 1) is a stationary stochastic Gaussian process. The plane of incidence is the x 1 x 3 plane, and the magnetic field is directed along the x 2-axis. In the presence of the magnetic field the dispersion curve for the surface polaritons supported by the surface in the absence of the random roughness becomes non-reciprocal, i.e. the wavenumber k +(ω) for a surface polariton of frequency ω propagating in the +x 1-direction is unequal to the (magnitude of the) wavenumber k −(ω) for a surface polariton of the same frequency propagating in the −x 1-direction. As a consequence of this they find that the peak in the angular distribution of the intensity of the incoherent com...

The transverse correlation length for randomly rough surfaces

It is shown by numerical simulations for a random, one-dimensional surface defined by the equationx3=ζ(x1), where the surface profile function ζ(x1) is a stationary, stochastic, Gaussian process, that the transverse correlation lengtha of the surface roughness is a good measure of the mean distance 〈d〉 between consecutive peaks and valleys on the surface. In the case that the surface height correlation function 〈ζ(x1)ζ(x1′)〉/〈ζ2(x1)〉=W (|x1−x1′|) has the Lorentzian formW(|x1|)=a2/(x 1 2 +a2) we find that 〈d〉=0.9080a; when it has the Gaussian formW(|x1|)=exp(−x 1 2 /a2), we find that 〈d〉=1.2837a; and when it has the nonmonotonic formW(|x1|)=sin(πx1/a)/(πx1/a), we find that 〈d〉=1.2883a. These results suggest that 〈d〉 is larger, the faster the surface structure factorg(|Q|) [the Fourier transform ofW(|x1|)] decays to zero with increasing |Q|. We also obtain the functionP(itx1), which is defined in such a way that, ifx1=0 is a zero of ζ′(x1),P(x1)dx1 is the probability that the nearest zero of ζ′(x1) for positivex1 lies betweenx1 andx1+dx1.

Enhanced backscattering of light from a non-Gaussian random metal surface

We study theoretically the scattering of a finite beam of p- or s-polarized light from a randomly rough metal surface defined by the equation x3 = ζ(x1), where the surface profile function ζ(x1) is not a Gaussian random process. The surface that we study is a parallel array of identical ridges and grooves, of Gaussian form, whose separations along the x1 axis are random, i.e., a random grating. First we show that the surface defined in this way is a non-Gaussian surface. A numerical algorithm based on the fast Fourier transform is then developed for constructing such surface profiles, and the probability-density functions for the height and slope of the surface at an arbitrary point are calculated. The angular distribution of the intensity of the incoherent component of the light scattered from an ensemble of such surfaces is then calculated by purely numerical methods for surfaces with large root-mean-square slope (=1). A well-defined peak in this angular distribution in the retroreflection direction—enhanced backscattering—is observed for both polarizations. It is shown that this enhanced backscattering is a multiple-scattering effect.

On the inversion of the atomic beam scattering data in the Eikonal approximation

The inversion method of Celli, Garcia and Rieder requires some a-priori knowledge about the phases of the dominant scattering amplitudes. A perturbational method will be introduced here which is evaluated in the Eikonal approximation, but can in principle be extended to the Helmholtz equation. It is based on the observation that the expansion coefficients P G ( n) of the scattering intensities with respect to k i, the incident wave vector, are functions of the corrugation amplitudes and phase differences and fulfill certain sum rules. If these coefficients are determined by several measurements with varying scattering parameters, then the lowest P G ( n) for n = 2,3.4 are sufficient for finding the essentially unique surface profile function. As an example, simulated surface domain structure scattering data will be inverted.

Surface topography in scanning tunneling microscopy: A free-electron model.

The topographic image as given by scanning tunneling microscopy (STM) is deduced in analytic form in a free-electron, or Sommerfeld model. The method is non-numerical and employs perturbed wave functions for an arbitrarily modified plane metal surface to approximate the local density of states (LDOS), at the Fermi level. The curves of constant LDOS, hence also the contours followed by the probe in an s-wave tip model, are calculated in terms of h${(\mathrm{x}}_{?}$), a surface profile function. The image of an arbitrary periodic or nonperiodic surface structure is determined by contours of the form z${(\mathrm{x}}_{?}$)=z\ifmmode\bar\else\textasciimacron\fi{}+\ensuremath{\Delta}${(\mathrm{x}}_{?}$,z\ifmmode\bar\else\textasciimacron\fi{}) where z\ifmmode\bar\else\textasciimacron\fi{} is the average probe-surface separation, and \ensuremath{\Delta}${(\mathrm{x}}_{?}$,z\ifmmode\bar\else\textasciimacron\fi{}) is a convolution over h${(\mathrm{x}}_{?}$). We also discuss the parallel and perpendicular resolution of surface structures such as a one- or two-dimensional Gaussian, a perfect step, and a cosine surface, as a function of distance and tip radius. We find there is considerable smoothing of the image in STM for finite surface defects for typical tip-surface separations and tip radii.

Influence of surface corrugations on electron-stimulated desorption: Angular distributions of ions and neutral atoms from Ni(110)-CO.

The conventional theory of electron-stimulated desorption is modified to include the effect of a surface corrugation via a surface profile function h(x,y). The method is used to interpret azimuthal asymmetry in recently observed angular distributions of ions and neutral atoms from Ni(110)-CO. The principal effect comes from modification of the Hagstrum exponential form for the neutralization rate to account for the nonplanar surface. The simple model R(x,y,z) =Aexp{-a[z-h(x,y)]}, which is valid for h(x,y) and \ensuremath{\nabla}h(x,y) small, is shown to work very well.

Polariton resonant absorption in a bigrating: Exact theory compared to recent experiments and to perturbation theory

A nonperturbative theory of light scattering from a bigrating, with resonant excitation of surface polaritons, has been implemented to fit the results of a recent experiment. The experiment measured light absorption in Ag, with results that showed some discrepancies with first-order perturbation theory — particularly at 2-polariton peaks. The exact theory has accounted for some but not all of the discrepancies, indicating that their cause is not entirely due to perturbation theory's underestimating higher-order processes. A comparison of the Fourier coefficients of the surface profile function as determined by the two theories indicates that perturbation theory may develop problems in dealing with the cross-terms in the expansion of 2-dimensional profiles, even for small corrugation strengths.

Some aspects of light scattering from a randomly rough metal surface

Two problems involving the interaction of a volume electromagnetic wave with a randomly rough metal surface are studied. In the first part a recently constructed theory of the resonant nonspecular scattering of light from a randomly rough metal surface is fitted to experimental data in a way that permits the extraction of the two-dimensional Fourier transform g(k∥) of the two-point correlation function of the surface profile function from the experimental results. It is found that the resulting correlation function can have its first maximum away from k∥ = 0 and decays to zero with increasing k∥ in a nonmonotonic fashion. This form for g(k∥) has been obtained in recent, independent, experimental determinations of this function. In the second part the scattering of p-polarized light from a randomly rough grating is studied in a case in which the plane of incidence is normal to the grooves of the grating. A diagrammatic method that self-consistently takes into account the diagrams responsible for localization phenomena in other contexts is used in this analysis. It is shown that the contribution from these diagrams gives rise to an enhanced scattered intensity in the antispecular direction. As a by-product of this calculation, the localization of surface polaritons by surface roughness is demonstrated and their localization length is determined.