Beckmann's Theory Research Articles

Fluctuation correlation of the scattered intensity from two-dimensional rough surfaces

The fourth-order moment of the scattered light, namely, the correlation function of the scattered intensity fluctuation from two-dimensional optically weak homogeneous and isotropic rough surfaces obeying Gaussian distribution are investigated based on Beckmann theory and Gaussian moment theorem. Analytical and numerical results are given for the correlation functions of the scattered intensity fluctuation. Also two important special cases, two-frequency correlation and angular correlation, are discussed, as well as the influence of the incident and observation conditions and the characteristic parameters of the rough surfaces on the correlation function which could lead to a more clear understanding of the scattering property of the rough surface and provide a theoretical basis for the 3D target recognition.

A model of optical reflection on rough surface and its application to Monte Carlo simulation of light transport

Photon transport is important for designing an optical system of scintillation counters or Cherenkov counters. Monte Carlo simulations are useful in predicting the characteristics, since they are flexible and cost us less. Optical reflection is an important part of such simulation studies. Light reflection on a rough surface is usually treated by Lambert's law that holds for reflection on an isotropically diffusive surface, while mirror reflection is the other extreme case. P. Beckmann's theory (1967) of reflection on rough surfaces lies in-between. A Monte Carlo program was developed based on this model. The programs were tested on the calculations of light collection efficiency and photon transport problems. The results were compared with those with pure mirror reflectors and that with isotropically diffusive reflectors.

The DBO method for scattering: extension to dielectric and metallic rough surfaces

An extension of the DBO (delta function boundary operator) method to the computation of the field scattered by dielectric or metallic rough surfaces is presented. As for perfectly conducting rough surfaces, the numerical implementation is achieved by using the technique of FFT, which leads to very short computation times. Comparisons with rigorous results and with Beckmann's theory (1967) lead to conclusions which depend on the conductivity of the material.

Experimental study of scattering from characterized random surfaces

An experimental investigation of light scattering from random rough surfaces is described. The surfaces, whose height fluctuations approximately follow Gaussian statistics, are fabricated in photoresist with a metal overcoating. When the lateral correlation length is larger than a wavelength and the surface slopes are mild, measurements of diffuse scattering are found to agree with the Beckmann theory, as long as the angle of incidence is not too large. For other surfaces that have stronger slopes, depolarization and enhanced backscattering may be observed in the diffuse scattering. Though we are unaware of theoretical calculations that compare with the measurements, the effects of multiple scattering are shown to be consistent with the major features of the observations.

Incorporation of intraocular scattering in schematic eye models.

Beckmann's theory of scattering from rough surfaces is applied to obtain, from the experimental veiling glare functions, a diffuser that when placed at the pupil plane would produce the same scattering halo as the ocular media. This equivalent diffuser is introduced in a schematic eye model, and its influence on the point-spread function and the modulation-transfer function of the eye is analyzed.

Determination de la longueur de covariance d'une surface rugueuse par l'etude de la diffusion residuelle a l'infini

We propose a simple method for directly measuring the correlation length of a slightly rough surface by using the residual scattered light. As in a previous work about strong diffusors, we use the experimental results to discuss the validity of Beckmann's theory on surface scattering, now in the case of weak diffusors.

Grating Efficiencies in the Soft X-ray Region a Comparison of Different Theories

A numerical study is presented of the application of two approximate grating theories, the Rayleigh theory and the Beckmann theory, in the soft X-ray region. Comparison with a rigorous electromagnetic theory makes it possible to estimate the wavelengths and angles of incidence for which these theories give accurate results. Mainly, gratings having a sinusoidal groove profile are studied, but results for lamellar and echelette gratings are also discussed.

A Detailed Study of the Scattering of Scalar Waves from Random Rough Surfaces

A multiple scattering theory of scalar waves from random rough surfaces is presented. By using the Ewald-Oseen extinction theorem the scattering integral equation is solved by means of an expansion in σ powers (σ being the standard deviation of the corrugation). Values of the mean scattered intensity until the fourth order of σ are given. The quick convergence of this series for low σ permits us to deal with those situations of small roughness in which the Kirchhoff approximation given by Beckmann and Spizzichino [2] fails. These are the cases in which σ ≳ T and λ ≳ T (λ being the wavelength and T the surface correlation length). Thus this theory can give the intensity for white noise surfaces, and yields the conditions under which the single scattering Kirchhoff approximation works, as well as its percentage of error. As such it is shown that Beckmann's theory gives good results in all cases in which σ /T < 0·05 and, thus, the reason why it is valid for interpreting laser speckle measurements is given. A...

Non-circular gaussian speckle contrast in the exact theory of multiple scattering of waves from random rough surfaces

In this paper the expression for the speckle contrast, under the assumption of non-circular gaussian statistics of the scattered field, is derived by using the exact scattering theory of waves by random rough surfaces previously established by the authors. This implies to find the second order moment 〈 A 2〈 of the scattering amplitude. It is shown that this zero outside the specular direction for large correlation distances of the random heights of the surface. as such, it coincides with the value obtained from the Kirchhoff approximation of Beckmann's theory and contributes to support the use of Beckmann's theory for measurements of speckle contrast in the specular direction, even when multiple scattering takes place.

Etudes des etats de surface par les proprietes de diffusion a l'infini en lumiere transmise

We apply Beckmann's theory on the reflection scattering of electromagnetic waves to the case of laser light transmitted by a ground glass. The variance of surface height and the correlation length were calculated and experimentally measured for different ground glasses. In order to obey the theoretical condition that the surface should be slightly rough, we immersed the ground glass surface in a liquid of suitable index of refraction. Our results show that the usual assumption about the statistical characteristics of rough surfaces do not apply to the surfaces we studied.

A comparison between talysurf 10 and optical measurements of roughness and surface slope

Talysurf 10 and specular and diffuse light reflection results are compared for a set of ground metal surfaces. The specular reflection results correlate well with Talysurf roughness values. The intensity-incidence variation can be regarded as the characteristic function of skewed roughness probability distributions of a type suggested by earlier work. One apparently anomalous result helps to show the usefulness of the optical methods. Adequate representation of the diffuse reflection results is achieved by a double sech function with two angular dispersion parameters. The Beckmann theory is applied to derive the autocorrelation function of the slopes which shows a correlation length of around 0.15 μm. This small value may explain the fact that the scattered light is dispersed through angles much greater than the mean slope indicated by the Talysurf. The inner dispersion parameter of the double sech function is closer to the Talysurf values but the significance of this is not clear.

Surface roughness measurements from coherent light scattering

Results are presented of surface roughness measurements obtained by applying Beckmann's theory on the scattering of electromagnetic waves from rough surfaces to the scattering of laser light from ground glass surfaces. Measurements of the variance of surface height obtained in several scattering geometries and also from stylus measurements are presented and shown to agree closely for each surface. Also results are presented which show that the usual assumption of a Gaussian form for the autocorrelation function of surface height does not apply to the surfaces being studied here.

A re-formulation of some results of P. Beckmann for scattering from rough surfaces

Beckmann's theory is restated in a form suitable for scattering in the optical region and some applications to the estimation of surface roughness are indicated.

Sound Scattering into the the Shadow Zone below an Isothermal Layer

A mathematical model is developed to describe sound propagation into the shadow zone below an isothermal layer by means of scattering from a rough surface. The model incorporates formula that describes the propagation losses within the duct to points from which the sound is scattered. The Beckmann theory is then used to estimate the scattered intensities at the various angles that correspond to ray paths leading from the surface to a point in the shadow zone. Reasonable agreement is found between the theoretical results and observations made at 5 kHz in a moderately rough sea. The theory indicates that most of the scattering occurs in a narrow region of the surface between 200 and 1200 yd from the point of observation in the shadow zone.

Light Scattering by Laser Mirrors

The Beckmann theory of scattering is applied to laser mirrors. It is shown that the theory correctly predicts the wavelength dependence of scattering profiles. The theory can be used to obtain numbers (sigma and Tau) for a laser mirror which describe its scattering characteristics. Scattering measurements made on eleven laser mirrors are presented.