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We show that the amplitude and phase information from a two-dimensional complex field can be synthesized from a phase-only optical element with micrometric resolution. The principle of the method is based on the combination of two spatially sampled phase elements by using a low-pass filter at the Fourier plane of a 4 - f optical system. The proposed encoding technique was theoretically demonstrated, as well as experimentally validated with the help of a phase-only spatial light modulator for phase encoding, a conventional CMOS camera to measure the amplitude of the complex field, and a Shack-Hartmann wavefront sensor to determine its phase. This work was funded by the Generalitat Valenciana through the programme (PROMETEO/2012/021), and by University Jaume I through the project P1·1B2013-53. The authors are also very grateful to the SCIC of the Universitat Jaume I for the use of the femtosecond laser.

Using the maximum and the minimum of interference, a novel two-step phase demodulation algorithm is proposed to perform the phase extraction in phase-shifting interferometry. By means of the simulation calculation and the experimental research, it is proved that both the measured phase and the phase shift with high precision can be obtained in the proposed algorithm.

A direct measurement of the amplitude and the phase of a femtosecond light pulse is performed for the first time to our knowledge. The measurement is made in the frequency domain, and the time dependence of the field can be easily obtained by a Fourier transform. The technique relies on a pulse synthesis scheme to unravel the frequency dependence of the phase. A mask filters the spectrum, which gives rise to a pulse with a measurable temporal profile related to the frequency dependence of the phase. In particular, with a rectangular slit the time delay of the synthesized pulse is the first derivative of the phase with respect to the frequency of the original pulse at the central frequency of the filter. The amplitude of the spectrum is obtained from the power spectrum.

The retrieval of phase information from only the magnitude of the Fourier transform of a signal remains an important problem for many applications. We present an algorithm for phase retrieval when there exist two related sets of Fourier-transform magnitude data. The data are assumed to come from a single object observed in two different polarizations through a distorting medium, so the phase component of the Fourier transform of the object is corrupted. Phase retrieval is accomplished by minimization of a suitable criterion function, which can take three different forms.

An experimental study of the use of an S-transform to evaluate the phase distribution from a two-dimensional fringe pattern by introducing the carrier frequencies in two spatial directions, x and y, is presented. The phase distribution is extracted from the optical fringe pattern by using an S-transform gradient and S-transform phase methods. The experimental result for the Fourier transform profilometry algorithm is compared with the results of the S-transform analysis.

A new iterative algorithm to be used to precisely reconstruct near-field distribution from an interferogram of a laser output generated by a cyclic radial-shearing interferometer is proposed. First, by use of a window function around the zero-frequency part of the Fourier transform of the interferogram and calculation of the inverse Fourier transform of the zero-frequency part, we obtain the background intensity distribution of the interferogram. Then, according to the iterative algorithm, the near-field distribution of the laser output is precisely reconstructed from the background intensity distribution obtained in the first step. A numerical simulation and an actual experiment of the near-field reconstruction of the laser output with arbitrary amplitude distribution are implemented successfully.

We present an analytical solution for the phase introduced by a phase-only spatial light modulator to generate far-field phase and amplitude distributions within a domain of interest. The solution is demonstrated experimentally and shown to enable excellent control of the far-field amplitude and phase.

An algorithm based on the Fourier transform method without phase unwrapping is proposed to measure the 3D profile of objects. Phase unwrapping is essential and time consuming in conventional fringe projection techniques. To avoid this step, the method utilizes a color fringe pattern to substitute a gray-level fringe pattern. Intensity information can be used to extract the wrapped phase, while color information can be used to calculate the phase shift range. Combining the two parts, real phase value can be obtained without phase unwrapping. Experiments are conducted to verify the feasibility of this method.

We present a new Fourier-based exact solution for deterministic phase unwrapping from experimental maps of wrapped phase in the presence of noise and phase vortices. This single-step approach has superior performance for images with high phase gradients or insufficient digital sampling approaching 2pi/pixel and therefore performs as a fast and practical solution for the phase-unwrapping problem for experimental applications in applied optics, physics, and medicine.

A wide range of interferometric techniques recover phase information that is mathematically wrapped on the interval (-pi, pi). Obtaining the true unwrapped phase is a longstanding problem. We present an algorithm that solves the phase unwrapping problem, using a combination of Fourier techniques. The execution time for our algorithm is equivalent to the computation time required for performing eight fast Fourier transforms and is stable against noise and residues present in the wrapped phase. We have extended the algorithm to handle data of arbitrary size. We expect the state of the art of existing interferometric applications, including the possibility for real-time phase recovery, to benefit from our algorithm.

The Fourier transform of a coherent optical image can be evaluated physically by use of a single lens plus free-space propagation, thereby providing the basis for the field of Fourier optics. I point out that one can similarly evaluate the discrete Fourier transform of a sampled or pixelated optical array physically by passing the discrete array amplitudes through a network of single-mode fibers or optical waveguides. A passive optical network that evaluates the fast Fourier transform of a coherent array can be fabricated by use of (N/2)log(2)[N] optical 3-dB couplers plus small added phase shifts. Implementing such networks in fiber or integrated optical form could provide the basis for a possible technology of fiber Fourier optics.