Polarimetric measurement of temporal coherence in electromagnetic light beams.

We present a method to determine the degree of temporal coherence of a quasimonochromatic vectorial light beam by polarimetric measurements. More specifically, we employ Michelson's interferometer in which the polarization Stokes parameters of the output (interference) beam are measured as a function of the time delay. Such a measurement enables us to deduce the magnitudes of the coherence (two-time) Stokes parameters, and hence the degree of coherence, of the input beam. Compared to existing methods the current technique has the benefits that neither optical elements in the arms of the interferometer nor visibility measurements are needed. The method is demonstrated with a laser diode and a filtered halogen source of various degrees of polarization.

Cross-spectral purity of the Stokes parameters in random nonstationary electromagnetic beams.

We consider cross-spectral purity in random nonstationary electromagnetic beams in terms of the Stokes parameters representing the spectral density and the spectral polarization state. We show that a Stokes parameter being cross-spectrally pure is consistent with the property that the corresponding normalized time-integrated coherence (two-point) Stokes parameter satisfies a certain reduction formula. The current analysis differs from the previous works on cross-spectral purity of nonstationary light beams such that the purity condition is in line with Mandel's original definition. In addition, in contrast to earlier works concerning the cross-spectral purity of the polarization-state Stokes parameters, intensity-normalized coherence Stokes parameters are applied. It is consequently found that in addition to separate spatial and temporal coherence factors the reduction formula contains a third factor that depends exclusively on polarization properties. We further show that cross-spectral purity implies a specific structure for electromagnetic spectral spatial correlations. The results of this work constitute foundational advances in the interference of random nonstationary vectorial light.

Three-dimensional polarization effects in optical tunneling.

We consider the three-dimensional (3D) polarimetric properties of an evanescent optical field excited in the gap of a double-prism system by a random plane wave. The analysis covers the case of frustrated total internal reflection (FTIR), i.e., optical tunneling, and relies on the characteristic decomposition of the 3×3 polarization matrix. We find in particular that, for any incident partially polarized plane wave, the evanescent field inside the gap is necessarily in a nonregular, genuine 3D polarization state. We also show that the 3D polarimetric properties of the field at the second boundary are sensitive to the changes of the gap width and that the relevant effects occur for the smaller widths when the angle of incidence of the plane wave becomes larger. The results of this work uncover new aspects of the polarimetric structure of genuine 3D evanescent fields and may find applications in near-field optics and surface nanophotonics.

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.

Probing coherence Stokes parameters of three-component light with nanoscatterers.

We establish a method to determine the spectral coherence Stokes parameters of a random three-component optical field via scattering by two dipolar nanoparticles. We show that measuring the intensity and polarization-state fringes of the scattered far field in three directions allows us to construct all nine coherence Stokes parameters at the dipoles. The method extends current nanoprobe techniques to detection of the spatial coherence of random light with arbitrary three-dimensional polarization structure.

Singular-value decomposition and electromagnetic coherence of optical beams.

We investigate the implications of the singular-value decomposition of the cross-spectral density (CSD) matrix to the description of electromagnetic spectral spatial coherence of stationary light beams. We show that in a transverse plane any CSD matrix can be expressed as a mixture of two CSD matrices corresponding to beams which are fully polarized but in general spatially partially coherent. The polarization and coherence structures of these constituent beams are specified, respectively, by the singular vectors and singular values of the full CSD matrix. It follows that vector-beam coherence, including the coherence Stokes parameters and the degree of coherence, can be formulated in terms of only two correlation functions. We further establish two-point analogs of the spectral and characteristic decompositions of the polarization matrix and show that in the case of a Hermitian CSD matrix their composition is specified by the so-called degree of cross-polarization.

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.

Complete coherence of random, nonstationary electromagnetic fields.

Despite a wide range of applications, the coherence theory of random, nonstationary (pulsed or otherwise) electromagnetic fields is far from complete. In this work, we show that full coherence of a nonstationary vectorial field at a pair of spatiospectral points is equivalent to the factorization of the cross-spectral density matrix, and full pointwise coherence over a spatial volume and spectral band leads to a factored cross-spectral density throughout the domain. We further show that in the latter case, the time-domain mutual coherence matrix factors in the spatiotemporal variables, and the field is temporally fully coherent throughout the volume. The results of this work justify that certain expressions of random pulsed electromagnetic beams appearing in the literature can be called coherent-mode representations.

Poincaré sphere of electromagnetic spatial coherence.

We introduce a Poincaré sphere construction for geometrical representation of the state of two-point spatial coherence in random electromagnetic (vectorial) beams. To this end, a novel descriptor of coherence is invoked, which shares some important mathematical properties with the polarization matrix and spans a new type of Stokes parameter space. The coherence Poincaré sphere emerges as a geometric interpretation of this novel formalism, which is developed for uniformly and nonuniformly fully polarized beams. The construction is extended to partially polarized beams as well and is demonstrated with a field having separable spatial coherence and polarization characteristics. At a single point, the coherence Poincaré sphere reduces to the conventional polarization Poincaré sphere for any state of partial polarization.

Fast calculation of tightly focused random electromagnetic beams: controlling the focal field by spatial coherence.

Focusing of a vectorial (electromagnetic) optical beam through a high numerical aperture can be investigated by means of the Richards-Wolf diffraction integral. However, such an integral extends from two-dimensional to four-dimensional, greatly increasing the computation time and therefore limiting the applicability, when light with decreased spatial coherence is considered. Here, we advance an effective protocol for the fast calculation of the statistical properties of a tightly focused field produced by a random electromagnetic beam with arbitrary state of spatial coherence and polarization. The novel method relies on a vectorial pseudo-mode representation and a fast algorithm of the wave-vector space Fourier transform. The procedure is demonstrated for several types of radially (fully) polarized but spatially partially coherent Schell-model beams. The simulations show that the computation time for obtaining the focal spectral density distribution with 512 × 512 spatial points for a low coherence beam is less than 100 seconds, while with the conventional quadruple Richards-Wolf integral more than 100 hours is required. The results further indicate that spatial coherence can be viewed as an effective degree of freedom to govern both the transverse and longitudinal components of a tightly focused field with potential applications in reverse shaping of focal fields and optical trapping control.

Interferometric imaging of reflective micro-objects in the presence of strong aberrations.

Some imaging techniques reduce the effect of optical aberrations either by detecting and actively compensating for them or by utilizing interferometry. A microscope based on a Mach-Zehnder interferometer has been recently introduced to allow obtaining sharp images of light-transmitting objects in the presence of strong aberrations. However, the method is not capable of imaging microstructures on opaque substrates. In this work, we use a Michelson interferometer to demonstrate imaging of reflecting and back-scattering objects on any substrate with micrometer-scale resolution. The system is remarkably insensitive to both deterministic and random aberrations that can completely destroy the object's intensity image.

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.

Blackbody far-field coherence.

We revisit the spectral coherence properties of far-field radiation emanating from an aperture in a blackbody cavity on the basis of Kirchhoff's boundary conditions. We point out that the far zone cross-spectral density matrix derived earlier in the literature by separately propagating all three aperture-field components does not show transversality of the field at nonparaxial directions. This is not the case when Luneburg's diffraction integrals are applied on the transverse source field components to determine the entire far field. We compare the electromagnetic degrees of coherence for the two methods and show that over important angular separations their values coincide with high accuracy. The results of this work and of others concerning the far-field intensity, polarization, and paraxial angular coherence are in full agreement.

Ghost polarimetry using Stokes correlations.

We present here a novel ghost polarimeter based on Stokes parameter correlations and a spatially incoherent classical source with adjustable polarization state and Gaussian statistics. The setup enables extracting the four amplitudes and three phase differences related to the spectral $ 2 \times 2 $2×2 complex Jones matrix of any transmissive polarization-sensitive object. Our work extends the ghost imaging methods from the traditional intensity correlation measurements to the detection of polarization state correlations.

Temporal coherence and polarization modulation of pulse trains by resonance gratings.

We analyze the effects of subwavelength-period resonance gratings on temporally partially coherent optical plane-wave pulse trains. The interaction of the grating with pulses is simulated with the Fourier modal method and finite-difference time-domain method whose performances are compared. Both TE and TM linearly polarized Gaussian Schell-model pulse trains are examined, and partial temporal coherence is modeled with the identical elementary-pulse method. The polarization-dependent response of the grating is seen to lead to significant variations in the average intensity, polarization properties, and degree of temporal coherence of the reflected (and transmitted) pulse trains when the coherence time and polarization state of the incident field are altered. As an important example, we demonstrate that a fully polarized incident pulse train can become partially polarized in grating reflection. The results find use in tailoring of random electromagnetic pulse trains.

Electromagnetic Schell-model beams with arbitrary complex correlation states.

A recently suggested technique for modeling of the complex coherence states of scalar stationary light beams [Opt. Lett.43, 4727 (2018)OPLEDP0146-959210.1364/OL.43.004727] is extended to the electromagnetic (EM) domain. It is shown that spatially varying phase profiles of the 2×2 source correlation matrix allow for fine, independent, two-dimensional spatial control of the three components of the polarization matrix in the far field. An EM coherence sorter based on the linear phases of the coherence matrix components is introduced which enables spatial separation of the three independent components of the far-field polarization matrix without affecting their initial distributions.

Polarimetric nonregularity of evanescent waves.

Three-dimensional polarization states of random light can be classified into regular and nonregular according to the structure of the related 3×3 polarization matrix. Here we show that any purely evanescent wave excited in total internal reflection of a partially polarized plane-wave field is always in a nonregular polarization state. The degree of nonregularity of such evanescent waves is also studied in terms of a recently advanced measure. Nonregular evanescent waves uncover new aspects of the polarimetric structure and dimensional character of electromagnetic near fields, with potential applications in nanoscale surface optics.

Intensity and spin anisotropy of three-dimensional polarization states.

Anisotropy is a natural feature of polarization states, and only fully random three-dimensional (3D) states exhibit complete isotropy. In general, differences between the strengths of the electric field components along the three orthogonal directions give rise to intensity anisotropy. Moreover, polarization states involve an average spin whose inherent vectorial nature constitutes a source of spin anisotropy. In this work, appropriate descriptors are identified to characterize quantitatively the levels of intensity anisotropy and spin anisotropy of a general 3D polarization state, leading to a novel interpretation for the degree of polarimetric purity as a measure describing the overall polarimetric anisotropy of a 3D optical field. The mathematical representation, as well as the physical features of completely intensity-isotropic 3D polarization states with a maximum spin anisotropy, are also examined. The results provide new insights into the polarimetric field structure of random 3D electromagnetic light states.

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.

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.