Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms

Abstract

Phase-space tomography is experimentally demonstrated for the determination of the spatially varying amplitude and phase of a quasi-monochromatic optical field by measurements of intensity only. Both fully and partially coherent sources are characterized. The method, which makes use of the fractional-order Fourier transform, also yields the Wigner distribution of the field and works in one or two dimensions.

Noninterferometric phase retrieval using a fractional Fourier system

“…Thus the optimal choice of ⌬␣ requires some a priori knowledge of the signal bandwidth and the noise in the system. Though this method requires only 2N samples as compared with N 2 samples required for methods that sample the entire phase space [12][13][14][15], more a priori knowledge of the signal is required, as compared with these latter methods, for the judicious choice of FRT orders ␣ 0 and ␣ 0 + ⌬␣.…”

“…In a broad sense, all of these methods extract the phase and thereby the complete signal information from single or multiple intensity measurements by using a deterministic algorithm based on the underlying physical model. Most of the deterministic phase retrieval methods are based on the transport of intensity model including the methods of Teague and Streibl. Another class of methods reconstructs the signal by sampling phase space distribution functions like the ambiguity function (AF) [12][13][14][15]. The underlying philosophy of these approaches, generically referred to as phase space tomography, is that the phase space distribution functions such as the AF contain the entire signal information.…”

“…In this case we can describe the two resulting FRT domains, into which the Wigner distribution function (WDF) of the signal has been projected, as being close, since their angular separation in phase space is small. Such a treatment helps us in comparing these methods with the phase space tomographic methods that are based on sampling the entire phase space [12][13][14][15]. From the literature it appears that the signal extraction methods based on output intensity measurements at two close FRT domains lead to a lesser number of necessary samples.…”

Programmable two-dimensional optical fractional Fourier processor

“…In particular, it is a useful tool for beam characterization, filtering, phase space tomography, phase retrieval, encryption, etc. [1,2]. The FRFT of a two-dimensional function f (r i ) for parameters γ x and γ y , which are known as transformation angles, is defined as…”

Direct measurement of the spatial Wigner function with area-integrated detection

“…First, it is based on a series of measurements performed with an area integrating detector that collects all available light power, rather than on spatially resolved detection employed is some of the previous proposals. [8,9] Therefore, it can be used in regimes where array detectors are not available, for example in mid-and far infrared, or when single photon signals are involved. In such cases, our technique provides the optimal signal-to-noise ratio, analogously to the case of Fourier transform spectroscopy which is more efficient than a spectrometer with a single scanning detector.…”