scholarly journals Characterization of Optical Aberrations Induced by Thermal Gradients and Vibrations via Zernike and Legendre Polynomials

2016 ◽  
Vol 06 (06) ◽  
pp. 113-123 ◽  
Author(s):  
Igor Di Varano
Keyword(s):  
Legendre Polynomials ◽  
Thermal Gradients ◽  
Optical Aberrations

On the circle polynomials of Zernike and related orthogonal sets

1954 ◽  
Vol 50 (1) ◽  
pp. 40-48 ◽  
Author(s):  
A. B. Bhatia ◽  
E. Wolf
Keyword(s):  
Unit Circle ◽  
Generating Functions ◽  
Legendre Polynomials ◽  
Phase Contrast ◽  
Diffraction Theory ◽  
Zernike Polynomials ◽  
Optical Aberrations ◽  
Orthogonal Set ◽  
Complete Orthogonal Set ◽  
Explicit Expressions

ABSTRACTThe paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with respect to rotations of axes about the origin of coordinates. It is found that though there exist an infinity of such sets there is only one set which in addition has certain simple properties strictly analogous to that of Legendre polynomials. This set is found to be identical with the set of the circle polynomials of Zernike which play an important part in the theory of phase contrast and in the Nijboer-Zernike diffraction theory of optical aberrations.The results make it possible to derive explicit expressions for the Zernike polynomials in a simple, systematic manner. The method employed may also be used to derive other orthogonal sets. One new set is investigated, and the generating functions for this set and for the Zernike polynomials are also given.

Generation and characterization of adjustable pure third-order spatial phase by tuning optical aberrations

2019 ◽  
Vol 21 (8) ◽  
pp. 085602 ◽  
Author(s):  
Daryoush Abdollahpour ◽  
Morteza Lotfollahi ◽  
Mohammad Yeganeh ◽  
Saifollah Rasouli
Keyword(s):  
Optical Aberrations ◽  
Third Order ◽  
Spatial Phase

An approach to characterization of resistance of brittle materials to thermal gradients

1999 ◽  
Vol 66 (7) ◽  
pp. 624
Author(s):  
A. V. Shatilov ◽  
G. T. Petrovski ◽  
I. A. Mironov
Keyword(s):  
Brittle Materials ◽  
Thermal Gradients

Thermal maturation of organic matter in the Middle Devonian to Tertiary section, Fort Norman area (central Mackenzie Plain)

1991 ◽  
Vol 28 (7) ◽  
pp. 1009-1018 ◽  
Author(s):  
S. Feinstein ◽  
G. K. Williams ◽  
L. R. Snowdon ◽  
F. Goodarzi ◽  
T. Gentzis
Keyword(s):  
Middle Devonian ◽  
Variable Temperature ◽  
Thermal Conditions ◽  
Thermal Gradients ◽  
Structural Pattern ◽  
Depositional History ◽  
Entire Study ◽  
Complex Structural ◽  
Entire Study Area

The Fort Norman area is marked by a complex structural and depositional history with temporally and spatially variable temperature and maturity gradients. Maturity of the Middle Devonian Canol – Hare Indian unit indicates two terrains of markedly different paleotemperatures and thermal gradients, roughly coinciding with the Palaeozoic Keele Arch (north) and Root Basin (south), with a sharp transition (lateral discontinuity) between them. Maturity of the unconformably overlying Cretaceous to early tertiary section is lower and laterally continuous across the entire study area, showing no relation to the sub-Cretaceous structures. The vertical maturity profile in the southern terrain is unique in that two discontinuous segments record markedly different thermal histories, neither of which is related to present burial conditions. Maturity of the Devonian formations was established some time prior to the sub-Cretaceous erosion, whereas that in the Cretaceous to Tertiary section postdates Palaeocene but predates a major part of the later Tertiary deformation and erosion. In contrast, the continuous maturity profile recorded in the northern terrain reflects a significant effect of the Tertiary thermal conditions, indicating much lower thermal gradients in the Keele Arch than in the Root Basin during the early phase of maturation. Along with the structural pattern and sedimentary history, the striking differences indicated in the thermal gradients between the Root Basin and the Keele Arch suggest a strike-slip-related pull-apart or another stretching mechanism for the Root Basin in contrast to the compression in the Keele Arch. In addition to the characterization of the regional thermal history and its variation with time, the maturity pattern obtained allows some estimate to be made of the thickness of the eroded section and timing of the major erosional phases.

ULTRASONIC CHARACTERIZATION OF THERMAL GRADIENTS IN A HOMOGENEOUS PLATE

2008 ◽  
Author(s):  
R. S. Martin ◽  
J. E. Michaels ◽  
T. E. Michaels ◽  
Donald O. Thompson ◽  
Dale E. Chimenti
Keyword(s):  
Thermal Gradients ◽  
Homogeneous Plate ◽  
Ultrasonic Characterization

Characterization of the optical aberrations of a metrological UV microscope

2021 ◽  
Author(s):  
Jan Krüger ◽  
Bernd Bodermann ◽  
Rainer Köning ◽  
Wolfgang Haessler-Grohne ◽  
Han Xu
Keyword(s):  
Optical Aberrations

The Description and Characterization of Symmetric Frames and Gabor Frames and Applications in Material Engineering

2013 ◽  
Vol 712-715 ◽  
pp. 2458-2463
Author(s):  
Qing Jiang Chen ◽  
Xiao Ting Lei ◽  
Jian Feng Zhou
Keyword(s):  
Legendre Polynomials ◽  
Materials Science ◽  
Tight Frame ◽  
Simple Approach ◽  
Gabor Frames ◽  
Wavelet Frames ◽  
Science And Engineering ◽  
Interdisciplinary Field ◽  
Frame Wavelets

Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. In this paper, we discuss a new set of symmetric tight frame wave-lets with the associated filterbanks outputs downsampled by several generators. The frames consist of several generators obtained from the lowpass filter using spectral factorization, with lowpass fil-ter via a simple approach using Legendre polynomials. The filters are feasible to be designed and offer smooth scaling functions and frame wavelets. We shall give an example to demonstrste that so -me examples of symmetric tight wavelet frames with three compactly supported real-valued sym- metric generators will be presented to illustrate the results.

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