Nonstationary optics: tutorial.

Over the past several decades, nonstationary optics has risen as a key enabling technology for a multitude of novel applications. These include areas of research such as micromachining and ultrafast optics, as well as the Nobel awarded research in femtochemistry, optical frequency combs, and attosecond physics. This tutorial aims to present some of the main concepts required to analyze nonstationary fields, with an emphasis on pulsed beams. The work begins from the fundamental building blocks of such fields, and builds up to some of their main properties. The spatiotemporal properties and stability of such fields are discussed in length, and some common measurement schemes are reviewed.

Reduction formula for cross-spectral purity of nonstationary light fields.

We examine cross-spectral purity of random, nonstationary (pulsed), scalar light fields with arbitrary spectral bandwidth. In particular, we derive a reduction formula in terms of time-integrated coherence functions, which ensures cross-spectral purity of interfering fields having identical normalized spectra. We further introduce fields that are cross-spectrally pure in either a global or local sense. Our analysis is based on an ideal field superposition realizable with all-reflective wavefront-shearing interferometers. Such devices avoid certain problems related to Young's interferometer, which is the framework customarily employed in assessing cross-spectral purity. We show that any partially coherent beam can be transformed into a locally cross-spectrally pure beam whose cross-spectral density is specular. On the other hand, lack of space-frequency (and space-time) coupling ensures cross-spectral purity in the global sense, i.e., across an entire transverse plane, regardless of the spectral bandwidth or the temporal shape of the pulses.

Randomness, determinism, and ignorance in coherence.

The effect of randomness and determinism on the coherence properties of light are studied in detail. As it is well known, a random field can have widely varying coherence properties. Here, it is shown that one can also produce a deterministic field with an arbitrarily low degree of coherence. The role of constant (non-random) fields are then considered, and some simulations with a toy model laser are presented. An interpretation of coherence as a measure of "ignorance" is presented.

Measurement of spatial coherence of light [Invited].

The most frequently used experimental techniques for measuring the spatial coherence properties of classical light fields in the space-frequency and space-time domains are reviewed and compared, with some attention to polarization effects. In addition to Young's classical two-pinhole experiment and several of its variations, we discuss methods that allow the determination of spatial coherence at higher data acquisition rates and also permit the characterization of lower-intensity light fields. These advantages are offered, in particular, by interferometric schemes that employ only beam splitters and reflective elements, and thereby also facilitate spatial coherence measurements of broadband fields.

Azimuthally periodic and radially quasi-periodic Bessel-correlated fields.

We introduce a class of partially coherent sources, which are capable of producing beams with radially quasi-periodic and azimuthally fully periodic intensity profiles. The physical properties of the source, as well as the propagation of the intensity distribution and the complex degree of spatial coherence of the ensuing beams are investigated and interpreted. It is shown that the shape and symmetry of the intensity and the degree of spatial coherence are generally adjustable and modulated by the parameters related to the beam source. Moreover, the periodic changes of intensity arise from the discontinuity of the phase. The results provide a method for synthesizing fields with peculiar periodic intensity distributions in polar coordinates.

Pulse shaping by spectral-domain polarization gratings.

We consider the spectral-domain counterparts of spatial-domain polarization gratings and study their effect on the temporal evolution of femtosecond-scale light pulses. These devices divide an incident light pulse to several orders via spectral polarization modulation, permitting pulse splitting and shaping with controlled time-domain polarization dynamics.

Propagation of Bessel-correlated specular and antispecular beams.

We address the specular properties of Bessel-correlated fields, generated by illuminating a tilted rotating plane-parallel glass plate with a coherent Gaussian beam and passing the output beam though a mirror-based wavefront folding interferometer. This device allows us to produce beams whose specular properties are preserved in propagation. In the far zone, the specular nature of these partially coherent fields is shown to produce intensity-profile oscillations in the sub-diffraction-limit scale. The analytical results at various propagation distances are verified experimentally by using another wavefront-folding interferometer for coherence measurements.

Coherence Switching with Metamaterials.

We demonstrate, theoretically, how the insertion of an enhanced epsilon-near-zero (EENZ) mirror in a laser cavity grants exceptional control over the coherence properties of the emitted light beam. By exploiting the peculiar sensitivity to polarization of EENZ materials, we achieve superior control over the spatial coherence of the emitted laser light, which can be switched at will between nearly incoherent and fully coherent, solely by means of polarization optics. Our EENZ cavity design is expected to be an efficient, compact, reconfigurable, and easily scalable source of light for illumination and speckle contrast imaging, as well as any other application that benefits from controlled spatial coherence.

Mirror-based scanning wavefront-folding interferometer for coherence measurements.

We demonstrate a modification to the traditional prism-based wavefront-folding interferometer that allows the measurement of spatial and temporal coherence, free of distortions and diffraction caused by the prism corners. In our modified system, the two prisms of the conventional system are replaced with six mirrors. The whole system is mounted on a linear XY-translation stage, with an additional linear stage in the horizontal arm. This system enables rapid and exact measurement of the full four-dimensional degree of coherence, even for relatively weak signals. The capabilities of our system are demonstrated by measuring the spatial coherence of two inhomogeneous and non-Schell model light sources with distinct characteristics.

Temporal coherence modulation of pulsed, scalar light with a Fabry-Pérot interferometer.

We analyze the effect of a high-finesse Fabry-Pérot interferometer on the temporal coherence properties of scalar optical plane-wave pulse trains. We focus on the cases of single-peak and double-peak transmissions of Gaussian Schell-model (GSM) and supercontinuum (SC) pulses. For the GSM light, we show how the characteristics of the average intensity and temporal degree of coherence of the transmitted pulses depend on the coherence parameters of the incident field. Regarding the SC light, the output is found to depend specifically on the location of the transmission peak(s) within the average spectrum. The results demonstrate that a Fabry-Pérot etalon can act as a simple passive element for tailoring the temporal (and spectral) coherence properties of optical pulse trains.

Scanning wavefront folding interferometers.

We present modified scanning-type wavefront folding interferometers (WFIs), which allow spatial coherence measurements of non-uniformly correlated fields, where the degree of coherence is a function of two absolute spatial coordinates instead of coordinate separation only (Schell model). As an alternative to conventional prism-based WFI implementations, we introduce a scheme based on reflections by three mirrors. This setup allows us to avoid obstructions due to prism corners, and it is remarkably robust to polarization effects. Experimental results on measurement of fields that do not obey the Schell model are provided with the three-mirror WFI, and the results are compared to those obtained with a Young's interferometer realized using a digital micromirror device.

Interferometry and coherence of nonstationary light.

We consider temporally integrating interferometric measurements and their relation to the coherence properties of nonstationary light. We find that performing such experiments as a function of time delay is equivalent to spectrally resolving the interference patterns, and time-domain coherence information can be obtained from field autocorrelation only if the source is of the Schell-model type. In an analogy to autocorrelation, we introduce field cross-correlation, which can be used to determine the complete complex field of unknown signal pulses if suitable probe pulses are available. We demonstrate our findings with simulated supercontinuum and free-electron laser ensembles, and discuss the prospect of carrying out experiments.

Complete spatial coherence characterization of quasi-random laser emission from dye doped transparent wood.

We report on the experimental determination of the complete two coordinate spatial coherence function of light emitted by a quasi-random laser, implemented on recently introduced dye-doped transparent wood. The spatial coherence was measured by means of a double grating interferometer, which has some advantages over the standard Young's interferometer. Analysis of the spatial coherence reveals that emission from such a material can be considered as a superposition of several spatial modes produced by individual emitters within semi-ordered scattering medium. The overall degree of coherence, γ¯, for this quasi-random laser was found to be 0.16 ± 0.01, having possible applications in speckle free laser imaging and illumination.

Paraxial propagation of a class of Bessel-correlated fields.

The propagation of a novel class of paraxial spatially partially coherent beams exhibiting Bessel-type correlations is studied in free space and in paraxial optical systems. We show that, under certain conditions, such beams can have functionally identical forms of the absolute value of the complex degree of spatial coherence not only at the source plane and in the far zone, but also at all finite propagation distances. Under these conditions the degree of spatial coherence properties of the field is a shape-invariant quantity, but the spatial intensity distribution is only approximately shape-invariant. The main properties of this class of model beams are demonstrated experimentally by passing a coherent Gaussian beam through a rotating wedge and measuring the coherence of the ensuing beams with a Young-type interferometer realized with a digital micromirror device.

Grating interferometer for light-efficient spatial coherence measurement of arbitrary sources.

We present a theoretical analysis and experimental verification of a z-scanning double-grating interferometer for spatial coherence measurements in space-frequency and space-time domains. This interferometer permits the measurement of spatial coherence between an arbitrary pair of points along a one-dimensional line, and in favorable conditions, it has a high light efficiency compared to the classical Young's two-pinhole experiment. The scheme is applicable to both quasi-monochromatic and broadband sources that need not obey the Schell model. We first provide experimental results with several narrowband primary and secondary sources, and then apply the technique to broadband sources with discrete and continuous spectra. In the latter case, the complex degree of (time-domain) spatial coherence is retrieved from spectrally resolved measurements using the Friberg-Wolf theorem [Opt. Lett.20, 623 (1995)OPLEDP0146-959210.1364/OL.20.000623]. We compare all results to those obtained with Young's interferometer realized using a digital micromirror device.

Self-focusing of a partially coherent beam with circular coherence.

In a recent publication [Opt. Lett.42, 1512 (2017)OPLEDP0146-959210.1364/OL.42.001512], a novel class of partially coherent sources with circular coherence was introduced. In this paper, we examine the propagation behavior of the spectral density and the spectral degree of spatial coherence of a beam generated by such a source in free space and in oceanic turbulent media. It is found that the beam exhibits self-focusing, which is dependent on the initial coherence and the parameters of oceanic turbulence. The self-focusing phenomenon disappears when the initial coherence is high enough or the oceanic turbulence is strong. The area of high coherence appears in the center and along two diagonal lines. With increasing turbulence, the coherence area reduces gradually along one diagonal line and is retained along the other one. A physical interpretation of the self-focusing phenomenon is presented, and potential applications in optical underwater communication and beam shaping are considered.

Bessel-correlated supercontinuum fields.

We examine the spatial coherence properties of supercontinuum fields generated by illuminating rotating bulk media with intense pulsed beams. Theoretical models are presented, which indicate the possibility of generating a class of Bessel-correlated fields (in time-averaged sense) using tilted plane-parallel glass plates and wedges as media for generation of supercontinuum radiation. In special cases, the ensuing fields have a strictly identical functional form in the spatial and angular domains. Some of the main results are verified experimentally by measuring the spatial coherence properties of bulk-generated supercontinuum fields using a wavefront-folding interferometer.

Derivation and validation of a novel Semi Empirical Deposition Estimation Model (SEDEM).

Predictive models are necessary in order to minimize potential damages in the event of a nuclear or radiological release. For this reason, a novel model for the calculation of both wet and dry deposition from airborne radioactivity is proposed. Full derivation of the model and the estimation of uncertainty are presented, and the validity of the model is evaluated by calculating deposition based on several measured airborne activities in different countries. The results are compared with the corresponding measured deposition activities and the predictive power of the model is found to be good, i.e. calculated depositions being within the limits of measurement uncertainty. Additionally, limitations of the model and possible sources of error in the calculations are discussed.

Transfer factors and effective half-lives of (134)Cs and (137)Cs in different environmental sample types obtained from Northern Finland: case Fukushima accident.

The Fukushima NPP accident caused a small but detectable cesium fallout in northern Finland, of the order of 1 Bq/m(2). This fallout transferred further to soil, water, flora and fauna. By using modern HPGe detector systems traces of (134)Cs from the Fukushima fallout were observed in various samples of biota. In northern Finland different types of environmental samples such as reindeer meat, berries, fish, lichens and wolf were collected during 2011-2013. The observed (134)Cs concentrations varied from 0.1 Bq/kg to a few Bq/kg. By using the known (134)Cs/(137)Cs ratio observed in Fukushima fallout the increase of the Fukushima accident to the (137)Cs concentrations was found to vary from 0.06 % to 6.9 % depending on the sample type. The aggregated transfer factors (Tag) and effective half-lives (Teff) for (134)Cs and (137)Cs were also determined and then compared with known values found from earlier studies which are calculated based on the fallout from the Chernobyl accident. Generally, the Tag and Teff values determined in this study were found to agree with the values found in the earlier studies. The Teff values were sample-type specific and were found to vary from 0.91 to 2.1 years for (134)Cs and the estimates for (137)Cs ranged between 1.6 and 19 years. Interestingly, the ground lichens had the longest Teff whereas the beard lichen had the shortest. In fauna, highest Tag values were determined for wolf meat ranging between 1.0 and 2.2 m(2)/kg. In flora, the highest Tag values were determined for beard lichens, ranging from 1.9 m(2)/kg to 3.5 m(2)/kg.