Almost-Fourier and almost-Fresnel transformations

Daniel F. V. James, G. S. Agarwal

Kurt Bernardo Wolf

Chisheng Li, Luo Han-Wen

Using the intermediate coordinate–momentum representation | x ⟩ s,r , we introduce a new Hadamard transform. It is found that the operator U corresponding to this transform can be considered as a combination of the Fresnel operator F ( r, s ) and the Fourier transform operator F by decomposing U. We also find that the matrix element s,r ⟨| x | U | f ⟩ just corresponds to an optical scaled Fresnel–Fourier transform.

Meng-Dan Zhao, Xu-Zhen Gao, Yue Pan +4 more

Abstract We present an optical encryption approach based on the combination of fractal Fresnel lens (FFL) and fractional Fourier transform (FrFT). Our encryption approach is in fact a four-fold encryption scheme, including the random phase encoding produced by the Gerchberg–Saxton algorithm, a FFL, and two FrFTs. A FFL is composed of a Sierpinski carpet fractal plate and a Fresnel zone plate. In our encryption approach, the security is enhanced due to the more expandable key spaces and the use of FFL overcomes the alignment problem of the optical axis in optical system. Only using the perfectly matched parameters of the FFL and the FrFT, the plaintext can be recovered well. We present an image encryption algorithm that from the ciphertext we can get two original images by the FrFT with two different phase distribution keys, obtained by performing 100 iterations between the two plaintext and ciphertext, respectively. We test the sensitivity of our approach to various parameters such as the wavelength of light, the focal length of FFL, and the fractional orders of FrFT. Our approach can resist various attacks.

Ajoy Kumar Ghatak, S. Lokanathan

Hong-Yi Fan, Lu Hai-Liang