scholarly journals Optical implementation of the two-dimensional fractional Fourier transform with different orders in the two dimensions

1995 ◽  
Vol 120 (3-4) ◽  
pp. 134-138 ◽  
Author(s):  
Aysegul Sahin ◽  
Haldun M. Ozaktas ◽  
David Mendlovic
Keyword(s):  
Fourier Transform ◽  
Fractional Fourier Transform ◽  
Two Dimensions ◽  
Two Dimensional

TWO-DIMENSIONAL NONLINEAR OPTICS SPECTROSCOPY: SIMULATIONS AND EXPERIMENTAL DEMONSTRATION

1996 ◽  
Vol 05 (03) ◽  
pp. 465-476 ◽  
Author(s):  
L. LEPETIT ◽  
G. CHÉRIAUX ◽  
M. JOFFRE
Keyword(s):  
Fourier Transform ◽  
Nonlinear Response ◽  
Two Dimensions ◽  
Second Order ◽  
Phase Matching ◽  
Two Dimensional ◽  
Experimental Demonstration ◽  
Spectral Interferometry ◽  
Dimensional Measurement ◽  
A New Technique

We propose a new technique, using femtosecond Fourier-transform spectral interferometry, to measure the second-order nonlinear response of a material in two dimensions of frequency. We show numerically the specific and unique information obtained from such a two-dimensional measurement. The technique is demonstrated by measuring the second-order phase-matching map of two non-resonant nonlinear crystals.

scholarly journals The Solvability of a Class of Convolution Equations Associated with 2D FRFT

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1928
Author(s):  
Zhen-Wei Li ◽  
Wen-Biao Gao ◽  
Bing-Zhao Li
Keyword(s):  
Fourier Transform ◽  
Convolution Operator ◽  
Fractional Fourier Transform ◽  
Hankel Transform ◽  
Polar Coordinates ◽  
Two Dimensional ◽  
Spatial Shift ◽  
Fractional Hankel Transform ◽  
The Relationship ◽  
Convolution Equations

In this paper, the solvability of a class of convolution equations is discussed by using two-dimensional (2D) fractional Fourier transform (FRFT) in polar coordinates. Firstly, we generalize the 2D FRFT to the polar coordinates setting. The relationship between 2D FRFT and fractional Hankel transform (FRHT) is derived. Secondly, the spatial shift and multiplication theorems for 2D FRFT are proposed by using this relationship. Thirdly, in order to analyze the solvability of the convolution equations, a novel convolution operator for 2D FRFT is proposed, and the corresponding convolution theorem is investigated. Finally, based on the proposed theorems, the solvability of the convolution equations is studied.

Numerical Simulation of Non-Gaussian Wave Elevation and Kinematics Based on Two-Dimensional Fourier Transform

2006 ◽  
Author(s):  
Xiang Yuan Zheng ◽  
Torgeir Moan ◽  
Ser Tong Quek
Keyword(s):  
Fourier Transform ◽  
Wave Interaction ◽  
Wave Theory ◽  
Two Dimensions ◽  
Second Order ◽  
Interaction Matrix ◽  
Random Wave ◽  
Two Dimensional ◽  
Near Surface ◽  
Wave Elevation

The one-dimensional Fast Fourier Transform (FFT) has been extensively applied to efficiently simulate Gaussian wave elevation and water particle kinematics. The actual sea elevation/kinematics exhibit non-Gaussianities that mathematically can be represented by the second-order random wave theory. The elevation/kinematics formulation contains double-summation frequency sum and difference terms which in computation make the dynamic analysis of offshore structural response prohibitive. This study aims at a direct and efficient two-dimensional FFT algorithm for simulating the frequency sum terms. For the frequency difference terms, inverse FFT and FFT are respectively implemented across the two dimensions of the wave interaction matrix. Given specified wave conditions, not only the wave elevation but kinematics and associated Morison force are simulated. Favorable agreements are achieved when the statistics of elevation/kinematics are compared with not only the empirical fits but the analytical solutions developed based on modified eigenvalue/eigenvector approach, while the computation effort is very limited. In addition, the stochastic analyses in both time-and frequency domains show that the near-surface Morison force and induced linear oscillator response exhibits stronger non-Gaussianities by involving the second-order wave effects.

Non-Gaussian Random Wave Simulation by Two-Dimensional Fourier Transform and Linear Oscillator Response to Morison Force

2007 ◽  
Vol 129 (4) ◽  
pp. 327-334 ◽  
Author(s):  
Xiang Yuan Zheng ◽  
Torgeir Moan ◽  
Ser Tong Quek
Keyword(s):  
Fourier Transform ◽  
Time Domain ◽  
Two Dimensions ◽  
Second Order ◽  
Linear Oscillator ◽  
Interaction Matrix ◽  
Random Wave ◽  
Two Dimensional ◽  
Near Surface ◽  
Non Gaussian

The one-dimensional fast Fourier transform (FFT) has been applied extensively to simulate Gaussian random wave elevations and water particle kinematics. The actual sea elevations/kinematics exhibit non-Gaussian characteristics that can be represented mathematically by a second-order random wave theory. The elevations/kinematics formulations contain frequency sum and difference terms that usually lead to expensive time-domain dynamic analyses of offshore structural responses. This study aims at a direct and efficient two-dimensional FFT algorithm for simulating the frequency sum terms. For the frequency-difference terms, inverse FFT and forward FFT are implemented, respectively, across the two dimensions of the wave interaction matrix. Given specified wave conditions, the statistics of simulated elevations/kinematics compare well with not only the empirical fits but also the analytical solutions based on a modified eigenvalue/eigenvector approach, while the computational effort of simulation is very economical. In addition, the stochastic analyses in both time domain and frequency domain show that, attributable to the second-order nonlinear wave effects, the near-surface Morison force and induced linear oscillator response are more non-Gaussian than those subjected to Gaussian random waves.

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