Triple correlation

Abstract: By electrical analysis of the output from a hot-wire anemometer, it is possible to measure rapidly and accurately all the quantities appearing in the theoretical equations for the decay of isotropic turbulence. The technique for these measurements is described, and possible extensions and limitations of the method are discussed. Measurements have been made of the second and fourth derivatives of the longitudinal double-velocity correlation at the origin, the third derivative of the longitudinal triple-velocity correlation at the origin, the statistical distribution in time of the velocity fluctuations, and the integral scale of turbulence.

Abstract: Triple correlations and their Fourier transforms, called bispectra, have properties desirable for image-sequence analysis and reconstruction. Specifically, the triple correlation of a two-dimensional sequence is shift invariant, vanishes for symmetric probability-distribution-function processes including Gaussian random processes of unknown covariance, and can be used to recover the original sequence uniquely to within a linear phase shift. Discrete analysis is carried out for a deterministic signal in an additive random-noise field. This approach yields discrete algorithms for implementation and permits explicit treatment of the additive noise. Recursive and least-squares fast-Fourier-transform-based algorithms for reconstructing a two-dimensional discrete Fourier transform from the bispectrum are reviewed in detail. While phase retrieval using the least-squares algorithm requires phase unwrapping, the recursive method requires a more simple correction. The bispectrum is applied to the estimate of a randomly translating or rotating object from a sequence of noisy images. The technique does not require solution of the correspondence problem, which is the primary advantage. The method works in low signal-to-noise-ratio cases, when conventional solutions to estimating the object correspondence may fail. Experimental results presented include application of the method to a sequence of infrared images.

Abstract: By measuring the triple correlation of an ultrashort laser pulse, it is possible to determine the temporal envelope of the pulse intensity. No assumptions on the analytic form of the pulse shape and no iterative algorithms are necessary. The method is extremely stable even for noisy data, and is a powerful tool for pulses that have a strong amplitude modulation. A multi-shot and a single-shot arrangement of the triple correlator has been realized. PACS: 42.60.By; 42.65.-k; 42.79.Fm Laser pulses as short as several femtoseconds can now be generated in many laboratories all over the world [1]. Since electronic devices, and even the fastest streak cameras, are too slow to measure the temporal evolution of these pulses, there has been an extensive search for novel measurement techniques over the past few years. All of these techniques rely on autocorrelation or crosscorrelation methods of a pulsed electric field, using various nonlinear effects such as second harmonic generation [2] or the optical Kerr effect [3]. In order to extract the temporal intensity, either an analytic pulse shape has to be assumed or iterative algorithms have to be used [4]. With a new experimental setup that uses a cascaded thirdorder nonlinear effect and introduces two independent delay times, background-free triple correlation of a femtosecond laser pulse has been measured in similar fashion to the technique proposed by Weber and Dandliker [5]. The method is different from the scheme proposed in [6], which relies on a single-step third-order nonlinear effect and on counterpropagating beams. The triple-correlation theory has been successfully applied in astronomy in order to increase the resolution of imaging systems [7, 8] and to measure the temporal correlation properties of lasers operating near their threshold [9]. In the late 1960s, it was shown that the third-order intensity correlation is sufficient to determine the time-dependent intensity of a laser pulse [10]. Following a scheme proposed in the mid-1980s [11], the intensity of a short laser pulse can be extracted without any assumptions about the pulse shape and using a recursive algorithm only. The measurement contains redundant information, and therefore the pulse intensity can be calculated by various paths, leading to the correct pulse shape even for very noisy data. Since the pulse intensity is obtained by a direct mathematical operation, this method is well suited for pulses with strong amplitude modulations. It has been proposed that such tailored pulses can, for example, be used in quantum control experiments [12].