scholarly journals Computation of Analytical Zoom Locus Using Padé Approximation

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 581 ◽  
Author(s):  
Kang Min Kim ◽  
Sun-Ho Choe ◽  
Jae-Myung Ryu ◽  
Hojong Choi
Keyword(s):  
Rational Function ◽  
Focal Length ◽  
Padé Approximation ◽  
Analytic Form ◽  
Zoom Lens ◽  
Lens System ◽  
Depth Of Focus ◽  
Pade Approximation ◽  
Conventional Calculation ◽  
Back Focal Length

When the number of lens groups is large, the zoom locus becomes complicated and thus cannot be determined by analytical means. By the conventional calculation method, it is possible to calculate the zoom locus only when a specific lens group is fixed or the number of lens groups is small. To solve this problem, we employed the Padé approximation to find the locus of each group of zoom lenses as an analytic form of a rational function consisting of the ratio of polynomials, programmed in MATLAB. The Padé approximation is obtained from the initial data of the locus of each lens group. Subsequently, we verify that the obtained locus of lens groups satisfies the effective focal length (EFL) and the back focal length (BFL). Afterwards, the Padé approximation was applied again to confirm that the error of BFL is within the depth of focus for all zoom positions. In this way, the zoom locus for each lens group of the optical system with many moving lens groups was obtained as an analytical rational function. The practicality of this method was verified by application to a complicated zoom lens system with five or more lens groups using preset patents.

Meniscus Shape and Optical Performance of a MEMS-Based Liquid Micro-Lens System

2008 ◽  
Author(s):  
Shong-Leih Lee ◽  
Chao-Fu Yang
Keyword(s):  
Electric Field ◽  
Laplace Equation ◽  
Focal Length ◽  
Optical Systems ◽  
Lens System ◽  
Meniscus Shape ◽  
Optical Aberration ◽  
Numerical Result ◽  
Micro Lens ◽  
Back Focal Length

It is very difficult to fabricate tunable optical systems having an aperture below 1000 micrometers with the conventional means on macroscopic scale. Krogmann et al. (J. Opt. A: Pure Appl. Opt. vol. 8, 2006, pp. S330-S336) presented a MEMS-based tunable liquid micro-lens system with an aperture of 300 micrometers. The system exhibited a tuning range of back focal length between 2.3 mm and infinity by using the electrowetting effect to change the contact angle of the meniscus shape on silicon with a voltage of 0−45V. However, a serious optical aberration was found in their lens system. In the present study, a numerical simulation is performed for this same physical configuration by solving the Young-Laplace equation on the interface of the lens liquid and the surrounding liquid. The resulting meniscus shape produces a back focal length that agrees with the experimental observation excellently. To eliminate the optical aberration, an electric field is applied on the system. The electric field alters the Young-Laplace equation and thus changes the meniscus shape and the quality of the lens system. The numerical result shows that the optical aberration of the lens system can be essentially eliminated when a proper electric field is applied.

Modeling of Rails Expressed using Pade Approximation Considering Frequency Dependent Effect

2017 ◽  
Vol 137 (2) ◽  
pp. 147-153
Author(s):  
Akinori Hori ◽  
Hiroki Tanaka ◽  
Yuichiro Hayakawa ◽  
Hiroshi Shida ◽  
Keiji Kawahara ◽  
...  
Keyword(s):  
Padé Approximation ◽  
Pade Approximation ◽  
Frequency Dependent ◽  
Dependent Effect

scholarly journals A New Approach of Asymmetric Homoclinic and Heteroclinic Orbits Construction in Several Typical Systems Based on the Undetermined Padé Approximation Method

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Jingjing Feng ◽  
Qichang Zhang ◽  
Wei Wang ◽  
Shuying Hao
Keyword(s):  
Dynamic Systems ◽  
Approximation Method ◽  
Padé Approximation ◽  
Global Bifurcation ◽  
Heteroclinic Orbits ◽  
Symmetric System ◽  
Pade Approximation ◽  
New Approach ◽  
Homoclinic And Heteroclinic Orbits ◽  
Single Orbit

In dynamic systems, some nonlinearities generate special connection problems of non-Z2symmetric homoclinic and heteroclinic orbits. Such orbits are important for analyzing problems of global bifurcation and chaos. In this paper, a general analytical method, based on the undetermined Padé approximation method, is proposed to construct non-Z2symmetric homoclinic and heteroclinic orbits which are affected by nonlinearity factors. Geometric and symmetrical characteristics of non-Z2heteroclinic orbits are analyzed in detail. An undetermined frequency coefficient and a corresponding new analytic expression are introduced to improve the accuracy of the orbit trajectory. The proposed method shows high precision results for the Nagumo system (one single orbit); general types of non-Z2symmetric nonlinear quintic systems (orbit with one cusp); and Z2symmetric system with high-order nonlinear terms (orbit with two cusps). Finally, numerical simulations are used to verify the techniques and demonstrate the enhanced efficiency and precision of the proposed method.

Long-offset moveout for VTI using Padé approximation

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. C219-C227 ◽  
Author(s):  
Hanjie Song ◽  
Yingjie Gao ◽  
Jinhai Zhang ◽  
Zhenxing Yao
Keyword(s):  
Padé Approximation ◽  
Pade Approximation ◽  
Practical Applications ◽  
Long Offset ◽  
Wide Range ◽  
Isotropic Media ◽  
Transversally Isotropic ◽  
Error Accumulation ◽  
Anisotropy Parameters ◽  
Vti Media

The approximation of normal moveout is essential for estimating the anisotropy parameters of the transversally isotropic media with vertical symmetry axis (VTI). We have approximated the long-offset moveout using the Padé approximation based on the higher order Taylor series coefficients for VTI media. For a given anellipticity parameter, we have the best accuracy when the numerator is one order higher than the denominator (i.e., [[Formula: see text]]); thus, we suggest using [4/3] and [7/6] orders for practical applications. A [7/6] Padé approximation can handle a much larger offset and stronger anellipticity parameter. We have further compared the relative traveltime errors between the Padé approximation and several approximations. Our method shows great superiority to most existing methods over a wide range of offset (normalized offset up to 2 or offset-to-depth ratio up to 4) and anellipticity parameter (0–0.5). The Padé approximation provides us with an attractive high-accuracy scheme with an error that is negligible within its convergence domain. This is important for reducing the error accumulation especially for deeper substructures.

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