Far-field Intensity Measurements Research Articles

Efficient ptychographic phase retrieval via a matrix-free Levenberg-Marquardt algorithm.

The phase retrieval problem, where one aims to recover a complex-valued image from far-field intensity measurements, is a classic problem encountered in a range of imaging applications. Modern phase retrieval approaches usually rely on gradient descent methods in a nonlinear minimization framework. Calculating closed-form gradients for use in these methods is tedious work, and formulating second order derivatives is even more laborious. Additionally, second order techniques often require the storage and inversion of large matrices of partial derivatives, with memory requirements that can be prohibitive for data-rich imaging modalities. We use a reverse-mode automatic differentiation (AD) framework to implement an efficient matrix-free version of the Levenberg-Marquardt (LM) algorithm, a longstanding method that finds popular use in nonlinear least-square minimization problems but which has seen little use in phase retrieval. Furthermore, we extend the basic LM algorithm so that it can be applied for more general constrained optimization problems (including phase retrieval problems) beyond just the least-square applications. Since we use AD, we only need to specify the physics-based forward model for a specific imaging application; the first and second-order derivative terms are calculated automatically through matrix-vector products, without explicitly forming the large Jacobian or Gauss-Newton matrices typically required for the LM method. We demonstrate that this algorithm can be used to solve both the unconstrained ptychographic object retrieval problem and the constrained "blind" ptychographic object and probe retrieval problems, under the popular Gaussian noise model as well as the Poisson noise model. We compare this algorithm to state-of-the-art first order ptychographic reconstruction methods to demonstrate empirically that this method outperforms best-in-class first-order methods: it provides excellent convergence guarantees with (in many cases) a superlinear rate of convergence, all with a computational cost comparable to, or lower than, the tested first-order algorithms.

Quantitative reconstruction of the complex-valued object based on complementary phase modulations

In the coherent diffraction imaging (CDI) techniques, a key point is to reconstruct the complex-valued object from the far-field intensity measurements, i.e., solving the phase retrieval problem. However, due to this ill-posed problem, traditional phase retrieval algorithms often encounter some problems associated with the iteration convergence. In this work, complementary phase modulations (CPM) are introduced to generate different far-field intensity measurements. The namely CPM-based method aims to find out the global optimal solution by imposing multi-dimensional constraints, including the diverse intensity images at the Fourier plane and the CPM at the object plane. It is proved by the numerical simulations and the optical experiments that the convergence speed and the recovery accuracy could be greatly improved. Furthermore, the shifting complementary phase modulations (SCPM)-based method is proposed by introducing more CPMs. The reconstruction performance is further improved even when the phase range is larger, and the support constraints are not required. In addition, the SCPM-based method is more robust to the Poisson noise. With the outstanding reconstruction performance, the CPM-based methods may be helpful to phase imaging in the application of visible-light microscopy and X-ray imaging.

Interscale mixing microscopy: numerically stable imaging of wavelength- scale objects with sub-wavelength resolution and far field measurements.

We present an imaging technique that allows the recovery of the profile of wavelength-scale objects with deep subwavelength resolution based on far-field intensity measurements. The approach, interscale mixing microscopy (IMM), relies on diffractive elements positioned in the near-field proximity of an object in order to scatter information carried by evanescent waves into propagating part of the spectrum. A combination of numerical solutions of Maxwell equations and nonlinear fitting is then used to recover the information about the object based on far-field intensity measurements. It is demonstrated that IMM has the potential to recover wavelength/20 features of wavelength-scale objects in the presence of 10% noise.

Quantifying the intrinsic amount of fabrication disorder in photonic-crystal waveguides from optical far-field intensity measurements

Residual disorder due to fabrication imperfections has important impact in nanophotonics where it may degrade device performance by increasing radiation loss or spontaneously trap light by Anderson localization. We propose and demonstrate experimentally a method of quantifying the intrinsic amount of disorder in state-of-the-art photonic-crystal waveguides from far-field measurements of the Anderson-localized modes. This is achieved by comparing the spectral range where Anderson localization is observed to numerical simulations, and the method offers sensitivity down to ≃1 nm.

Spatially resolved measurement of femtosecond laser induced refractive index changes in transparent materials

We present a practical method to determine femtosecond laser induced refractive index changes in transparent materials. Based on an iterative Fourier transform algorithm, this technique spatially resolves the refractive index of complex structures by combining the dimensions of the modified region with the corresponding phase change extracted from far-field intensity measurements. This approach is used to characterize optical waveguides written by a femtosecond laser in borosilicate glass.

Measurement of hard x-ray lens wavefront aberrations using phase retrieval

Measuring the deviation of a wavefront from a sphere provides valuable feedback on lens alignment and manufacturing errors. We demonstrate that these aberrations can be accurately measured at hard x-ray wavelengths, from far-field intensity measurements, using phase retrieval with a moveable structure in the beam path. We induce aberrations on a hard x-ray kinoform lens through deliberate misalignment and show that the reconstructed wavefronts are in good agreement with numerical simulations. Reconstructions from independent data, with the structure at different longitudinal positions and significantly separated from the beam focus, agreed with a root mean squared error of 0.006 waves.

Analytical technique for subwavelength far field imaging

We develop an analytical technique for retrieving the size and shape of subwavelength objects using far-field measurements. The approach relies on subwavelength diffraction gratings scattering evanescent information into the far field along with a numerical algorithm that is capable of deconvoluting this information based on the far-field intensity measurements. Several examples are presented, demonstrating resolution on the order of λ0/20. The developed method can be used at any frequency range, and may become a practical alternative to scanning near-field microscopy.

One-dimensional hard x-ray field retrieval using a moveable structure

We present a technique that allows measuring the field of an x-ray line focus using far-field intensity measurements only. One-dimensional phase retrieval with transverse translation diversity is used to recover a hard x-ray beam focused by a compound kinoform lens. The reconstruction is found to be in good agreement with independent knife-edge scan measurements taken at separated planes. The approach avoids the need for measuring the beam profile at focus and allows narrower beams to be measured than the traditional knife-edge scan.

Phase retrieval with transverse translation diversity: a nonlinear optimization approach

We develop and test a nonlinear optimization algorithm for solving the problem of phase retrieval with transverse translation diversity, where the diverse far-field intensity measurements are taken after translating the object relative to a known illumination pattern. Analytical expressions for the gradient of a squared-error metric with respect to the object, illumination and translations allow joint optimization of the object and system parameters. This approach achieves superior reconstructions, with respect to a previously reported technique [H. M. L. Faulkner and J. M. Rodenburg, Phys. Rev. Lett. 93, 023903 (2004)], when the system parameters are inaccurately known or in the presence of noise. Applicability of this method for samples that are smaller than the illumination pattern is explored.

Variable Coherence Scattering Microscopy

We propose and demonstrate the feasibility of a new microscopic technique, which is based on variable coherence illumination. By manipulating the spatial coherence properties of an incident evanescent field, subwavelength resolution is achieved over a large field of view from far-field intensity measurements.

Light scarring in an optical fiber.

We report the first experimental study of wave scarring in an optical fiber with a noncircular cross section. This optical multimode fiber serves as a powerful tool to image waves in a system where light rays exhibit a chaotic dynamics. Far-field intensity measurements are used to provide a better identification of scars in the Fourier domain. This first experimental characterization of scarring effect in optics demonstrates the relevance of such an optical waveguide for novel experiments in wave chaos.

Low-threshold 7.3 μm quantum cascade lasers grown by gas-source molecular beam epitaxy

We report low-threshold 7.3 μm superlattice-based quantum cascade lasers. The threshold current density is 3.4 kA/cm2 at 300 K and 1.25 kA/cm2 at 79 K in pulsed mode for narrow (∼20 μm), 2-mm-long laser diodes. The characteristic temperature (T0) is 210 K. The slope efficiencies are 153 and 650 mW/A at 300 and 100 K, respectively. Power output is in excess of 100 mW at 300 K. Laser far-field intensity measurements give divergence angles of 64° and 29° in the growth direction and in the plane of the quantum wells, respectively. Far-field simulations show excellent agreement with the measured results.

Transverse mode analysis of a laser beam by near- and far-field intensity measurements

A quantitative measurement of laser-beam quality can be performed by determination of the presence of multiple transverse modes of the laser oscillator and by calculation of their power content. Along this line of argument, we discuss a new approach that, starting from near-field and far-field intensity measurements, can evaluate the complex excitation coefficients of the transverse modes in a laser beam. The exploitation of near-field measurements sharply improves the performances of the technique in those cases in which only far-field measurements are used. The validity of the method is confirmed by several accurate numerical simulations and by some experimental results relative to a multimode Q-switched Nd:YAG laser.

Simultaneous reconstruction of object and aperture functions from multiple far-field intensity measurements

The general problem of reconstructing two unknown functions from a number of measurements of the intensity of the Fourier transform of the product of the functions, where the functions are displaced relative to one another by a different amount for each measurement, is examined. It is shown that this problem is equivalent to a blind deconvolution problem, which suggests that a unique solution exists for functions of any number of dimensions. Such a problem arises when attempts are made to recover the specimen and probe functions from microdiffraction plane intensity measurements in a scanning transmission electron microscope. An iterative algorithm capable of recovering these functions is described, and examples of the operation of the algorithm are presented for simulated data involving one- and two-dimensional functions.

Characterization of the transverse modes in a laser beam: analysis and application to a Q-switched Nd:YAG laser

We have developed a numerical code that, starting from far-field intensity measurements, is able to evaluate the excitation coefficients of the transverse modes in a laser system.Both the coherent and incoherent mode cases are addressed, and, while the incoherent case is shown to be equivalent to a linear problem, the coherent case is discussed through its equivalence to the phase-retrieval problem. Problems arising from both ill posedness and the nonlinearity are discussed in detail.The validity of our approach is confirmed by several numerical simulations and some experimental results on the characterization of a Q-switched Nd:YAG laser.

Phase retrieval from experimental far-field intensity data

We demonstrate the reconstruction of real and positive objects from experimental far-field intensity measurements by means of two phase-retrieval algorithms. Both the iterative Fourier transform and the simulated annealing algorithms are used, and an analysis is made of the advantages and disadvantages of each of these procedures and also of combinations of both methods. The objects tested either were binary or had many gray levels. We worked with data with a considerable amount of experimental noise, and in addition we recognized the importance of taking into account the nonrandom distortions produced by the detecting devices, which can critically bias the results toward erroneous estimates of the objects. Noisy data, however, can create ambiguities in the reconstructions, and hence additional information may be needed to overcome this disadvantage.