scholarly journals Variational Theory of Reflection

1985 ◽  
Vol 38 (2) ◽  
pp. 113 ◽  
Author(s):  
John Lekner
Keyword(s):  
Transverse Electric ◽  
Grazing Incidence ◽  
Interface Thickness ◽  
Variational Theory ◽  
Variational Estimate ◽  
First Order ◽  
Reflection Amplitude ◽  
Scattering Phase ◽  
Finite Barrier ◽  
Transverse Electric Wave

Schwinger's variational method for the scattering phase shift produced by a central potential is adapted to reflection by a planar potential barrier (or well). The formulation is general, for an arbitrary transition between any two media, but the application here is limited to reflection at a barrier between media of equal potential energy. The simplest variational estimate for the reflection amplitude correctly tends to -1 at grazing incidence, as it must for any finite barrier. This is in contrast to the first order perturbation reflection amplitude, which diverges at grazing incidence. The same variational estimate is also correct to second order in the ratio of the interface thickness to the wavelength of the incident wave. The theory applies also to the reflection of the electromagnetic s (or transverse electric) wave at an interface between two media.

Estimating the error in variational scattering calculations

1978 ◽  
Vol 56 (10) ◽  
pp. 1358-1364 ◽  
Author(s):  
J. W. Darewych ◽  
R. Pooran
Keyword(s):  
Wave Scattering ◽  
Exponential Potential ◽  
S Wave ◽  
Order Error ◽  
Absolute Value ◽  
First Order ◽  
Scattering Phase ◽  
Square Well ◽  
Scattering Phase Shifts ◽  
Made In

We derive bounds to the absolute value of the error that is made in variational estimates of scattering phase shifts. These bounds, like the variational estimates, are second order in 'small' quantities and are, in this respect, an improvement on similar but first-order error bounds derived previously by Bardsley, Gerjuoy, and Sukumar. The s-wave scattering by a square well potential, in the Born approximation, and by an exponential potential, using a many parameter trial function, are used to illustrate the results.

Application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide-angle wave equation

Geophysics ◽  
2014 ◽  
Vol 79 (6) ◽  
pp. T313-T321 ◽  
Author(s):  
Hanming Chen ◽  
Hui Zhou ◽  
Yanqi Li
Keyword(s):  
Wave Equation ◽  
Computational Cost ◽  
Grazing Incidence ◽  
Equation System ◽  
Perfectly Matched Layer ◽  
Staggered Grid ◽  
Complex Frequency ◽  
Wide Angle ◽  
First Order ◽  
General Complex

A classical split perfectly matched layer (PML) method has recently been applied to the scalar arbitrarily wide-angle wave equation (AWWE) in terms of displacement. However, the classical split PML obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence. Our goal was to improve the computational efficiency of AWWE and to enhance the suppression of edge reflections by applying a convolutional PML (CPML). We reformulated the original AWWE as a first-order formulation and incorporated the CPML with a general complex frequency shifted stretching operator into the renewed formulation. A staggered-grid finite-difference (FD) method was adopted to discretize the first-order equation system. For wavefield depth continuation, the first-order AWWE with the CPML saved memory compared with the original second-order AWWE with the conventional split PML. With the help of numerical examples, we verified the correctness of the staggered-grid FD method and concluded that the CPML can efficiently absorb evanescent and propagating waves.

Inorganic mixed phase templated by a fatty acid monolayer at the air–water interface: the Mn and Mg case

2018 ◽  
Vol 20 (9) ◽  
pp. 6629-6637 ◽  
Author(s):  
Alae El Haitami ◽  
Michel Goldmann ◽  
Philippe Fontaine ◽  
Marie-Claude Fauré ◽  
Sophie Cantin
Keyword(s):  
Grazing Incidence ◽  
Water Interface ◽  
X Ray Diffraction ◽  
Inorganic Layer ◽  
X Ray ◽  
First Order ◽  
First Order Phase Transition ◽  
Air Water Interface ◽  
First Order Phase

A first-order phase transition with a peculiar feature is evidenced by means of in situ grazing incidence X-ray diffraction in the 2D organic phase-mediated nucleation of an inorganic layer.

Theory & Performance of Holographically-Formed Stigmatic X-Ray Gratings

1975 ◽  
Vol 19 ◽  
pp. 643-656
Author(s):  
R. J. Speer
Keyword(s):  
Fringe Pattern ◽  
Optical Design ◽  
Grazing Incidence ◽  
Laser Fusion ◽  
Molecular Chemistry ◽  
X Ray ◽  
First Order ◽  
Imaging Theory ◽  
Reflection Grating ◽  
Aberration Corrected

Holographically-formed X-ray reflection grating scan now be constructed with competitive groove efficiency to classically ruled types, down to a short wavelength diffracting limit of several angstroms.The gratings can be generated on any surface capable of intersecting the interference fringe pattern without shadowing. This fact alone brings several new X-ray optical design possibilities within reach, for example, by combining plane construction waves with steep aspheric substrates. The first order imaging theory of the grazing incidence mounting is discussed and compared to measured performance for aberration corrected stigmatic types.A new 5 meter spectrograph has also been constructed with applications in molecular chemistry, laser fusion research and synchro tronspectroscopy. The unit is available with fully prefocussed holographic and classical X-ray grating optics, and uses the precision miniature camera principle of fully interchangeable lenses, but applied, in this case to the entrance slit, grating and detector modules.

Dual-function splitting of the embedded grating with connecting layer

2019 ◽  
Vol 33 (11) ◽  
pp. 1850129
Author(s):  
Wenhua Zhu ◽  
Bo Wang ◽  
Chenhao Gao ◽  
Kunhua Wen ◽  
Ziming Meng ◽  
...  
Keyword(s):  
Transverse Electric ◽  
Groove Depth ◽  
Transverse Magnetic ◽  
Dual Function ◽  
First Order ◽  
Rigorous Coupled Wave Analysis ◽  
Rigorous Coupled Wave ◽  
Plane Light ◽  
High Diffraction ◽  
Modal Method

We design and optimize the embedded dual-function grating with connecting layer in Littrow mounting. By using modal method and rigorous coupled-wave analysis, grating parameters are analyzed and calculated including grating groove depth, thickness of connecting layer, and so on. The grating device can diffract the transverse electric-polarized plane light mainly in the first-diffractive order with high-diffraction efficiency of 98.36%. Meanwhile, for the transverse magnetic-polarized plane light, the diffraction efficiencies in the zeroth-order and the first-order corresponding to 49.34% and 49.29% are obtained, respectively.

Advance in NPL X-Ray Gratings and Spectrometers

1975 ◽  
Vol 19 ◽  
pp. 571-575
Author(s):  
M. Stedman ◽  
A. Franks
Keyword(s):  
Wavelength Range ◽  
Short Wavelength ◽  
Grazing Incidence ◽  
Vitreous Silica ◽  
Holographic Grating ◽  
X Ray ◽  
First Order ◽  
Rectangular Profile ◽  
20 Nm

NPL x-ray gratings for use at grazing incidence in the wavelength range of 0.05 to 20 nm have diffraction efficiencies in the first order ranging from approximately 3% at 0.1 nm to 20% at wavelengths greater than 2 nm. X-ray gratings may have either a rectangular profile produced by processing a ruled or holographic grating or may have a shallow blazed waveform. In both types of grating, the grooves are formed in a vitreous silica blank and are usually overcoated with gold.A precision short wavelength (0.05 to 0.5 nm) Rowland circle focusing spectrometer has bee n constructed, which can cater for concave gratings of radii between 5 and 15 m.

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