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Optics Letters

Mother wavelets for complex wavelet transform derived by Einstein-Podolsky-Rosen entangled state representation

Hong-Yi Fan, Lu Hai-Liang

Published Feb 2, 2007• 20 citations
Publisher

Abstract

The Einstein-Podolsky-Rosen entangled state representation is applied to studying the admissibility condition of mother wavelets for complex wavelet transforms, which leads to a family of new mother wavelets. Mother wavelets thus are classified as the Hermite-Gaussian type for real wavelet transforms and the Laguerre-Gaussian type for the complex case.

DOI
10.1364/ol.32.000554

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The Einstein-Podolsky-Rosen entangled state representation is applied to studying the admissibility condition of mother wavelets for complex wavelet transforms, which leads to a family of new mother wavelets. Mother wavelets thus are classified as the Hermite-Gaussian type for real wavelet transforms and the Laguerre-Gaussian type for the complex case.
Based on the quantum mechanics representation, we propose a novel 3D entangled wavelet transformation. This kind of three entangled state representation can be generated theoretically by using a two-cascaded beam splitter, which has the completeness and orthogonal property. Some properties of 3D WTs are discussed, including the Parseval theory of 3D WTs, its inverse formula, and the orthogonality of the mother wavelets in parameter space. In addition, the mother wavelet condition for the 3D WTs is also derived.
In this paper, we propose the so-called continuous Fresnel-wavelet combinatorial transform which means that the mother wavelet undergoes the Fresnel transformation. This motivation can let the mother-wavelet-state itself vary from |〉 to F†r,s|〉, except for variation within the family of dilations and translations. The Parseval's equality, admissibility condition and inverse transform of this continuous Fresnel-wavelet combinatorial transform are analysed. By taking certain parameters and using the admissibility condition of this continuous Fresnel-wavelet combinatorial transform, we obtain some mother wavelets. A comparison between the newly found mother wavelets is presented.