Tracking the space bandwidth product in optical systems

Using matrix based techniques it is shown how the Space-Bandwidth Product (SBP) of a signal, as indicated by the location of the signal energy in the Wigner Distribution Function (WDF), can be tracked through any quadratic phase optical system whose operation is described by the Linear Canonical Transform (LCT). We extend this to include offset optical elements and systems, which are described by a more general transform-the Special Affine Fourier Transform (SAFT) or Offset LCT. It is shown that by decomposing the overall system matrix associated with the SAFT, into a product of the matrices representing the individual optical components, the spatial extent, its location and the signals bandwidth can be determined at any point in the optical system. This approach has application in many areas of optical signal processing; for example, numerical simulation, optical and digital compression, speckle metrology, optical encryption, optical filtering and the recording of processed images. The method is shown to be a simple tool of fundamental importance in the context of the entropy of the propagated signal and by tracking the Space Bandwidth product we can determine the quantity of information according to the Nyquist sampling theorem. We also discuss future work in relation to superresolution.

[1]  H. Ozaktas,et al.  Fractional Fourier transforms and their optical implementation. II , 1993 .

[2]  John T. Sheridan,et al.  Image encryption and the fractional Fourier transform , 2003, SPIE OPTO-Ireland.

[3]  J. Sheridan,et al.  Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation an operator approach , 1994 .

[4]  Mj Martin Bastiaans.  Application of the Wigner distribution function in optics , 1997 .

[5]  Soo-Chang Pei,et al.  Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms. , 2003, Journal of the Optical Society of America. A, Optics, image science, and vision.

[6]  V. Namias.  The Fractional Order Fourier Transform and its Application to Quantum Mechanics , 1980 .

[7]  B Y Gu,et al.  Recursive algorithm for phase retrieval in the fractional fourier transform domain. , 1998, Applied optics.

[8]  Z. Zalevsky,et al.  The Fractional Fourier Transform: with Applications in Optics and Signal Processing , 2001 .

[9]  Kehar Singh,et al.  Optical encryption using a localized fractional Fourier transform , 2003 .

[10]  H. Ozaktas,et al.  Fractional Fourier transforms and their optical implementation. II , 1993 .

[11]  John T. Sheridan,et al.  Fractional Fourier transform-based image encryption: phase retrieval algorithm , 2003 .

[12]  B Javidi,et al.  Encrypted optical memory system using three-dimensional keys in the Fresnel domain. , 1999, Optics letters.

[13]  D Mendlovic,et al.  Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain. , 1996, Optics letters.

[14]  Jingjuan Zhang,et al.  A lensless optical security system based on computer-generated phase only masks , 2004 .

[15]  L. Z. Cai,et al.  Optical implementation of scale invariant fractional Fourier transform of continuously variable orders with a two-lens system , 2002 .

[16]  John T. Sheridan,et al.  Optical operations on wave functions as the Abelian subgroups of the special affine Fourier transformation. , 1994, Optics letters.

[17]  John T. Sheridan,et al.  Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms. , 2005, Journal of the Optical Society of America. A, Optics, image science, and vision.

[18]  Levent Onural,et al.  Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms , 1994 .

[19]  B. Dong,et al.  Beam shaping in the fractional Fourier transform domain , 1998 .

[20]  B Javidi,et al.  Optical image encryption based on input plane and Fourier plane random encoding. , 1995, Optics letters.

[21]  Kehar Singh,et al.  Optical encryption using quadratic phase systems , 2001 .

[22]  Kehar Singh,et al.  Double random fractional Fourier domain encoding for optical security , 2000 .

[23]  A. Lohmann.  Image rotation, Wigner rotation, and the fractional Fourier transform , 1993 .

[24]  John T. Sheridan,et al.  Optical image encryption by random shifting in fractional Fourier domains. , 2003, Optics letters.

[25]  Zeev Zalevsky,et al.  Space–bandwidth product of optical signals and systems , 1996 .

[26]  J. Goodman.  Introduction to Fourier optics , 1969 .

[27]  F. H. Kerr,et al.  On Namias's fractional Fourier transforms , 1987 .

[28]  J. Mayer,et al.  On the Quantum Correction for Thermodynamic Equilibrium , 1947 .

[29]  S. A. Collins.  Lens-System Diffraction Integral Written in Terms of Matrix Optics , 1970 .

[30]  Mj Martin Bastiaans.  Wigner distribution function and its application to first-order optics , 1979 .