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By using the recently introduced parametrization of an n-dimensional density matrix in terms of the indices of population asymmetry and the intrinsic coherences, we define descriptors in both integer and continuous forms of the effective dimension that take place for a complete description of a density matrix, thus providing accurate information beyond the rank of the density matrix. The concepts of dimensional folding, hidden dimensional purity, and dimensional entropy are introduced and discussed in view of the new approach presented. The results are applicable to any physical system represented by a density matrix, such as n-level quantum systems, qutrits, sets of interacting pencils of radiation, classical polarization states, and to transformations of density matrices, as occurs with quantum channels.

We propose a method of measuring the spatial coherence of light by means of a temporally modulated nonredundant slit array implemented on a digital micromirror device. We first formulate the theory of the spatial coherence measurement to incorporate a general case when the observation plane is not necessarily placed in the far field of the slit array. We then demonstrate experimentally that a single measurement determines the spatial coherence for 15 different slit separations accurately, even if background light is unavoidable, under the condition that a nonredundant array of six slits is illuminated evenly. These results clearly show that fast and highly reliable spatial coherence measurement is achievable with the proposed method without any difficulties.

We consider two closely spaced slits of subwavelength width, pierced in a metal screen, that are illuminated by short, arbitrarily polarized, plane-wave pulses. The p-polarized component generates plasmonic pulses propagating on the metal surface between the slits. In addition to the primary transmitted pulses, scattering of the plasmonic pulses by the opposite slit is found to produce secondary (echo) radiation pulses, which causes splitting, shaping, and time-dependent polarization modulation of the radiated pulses in the far field. We study these effects in the linear regime. A previously explored phenomenological model is extended to the time domain and is validated by rigorous numerical calculations. Our analytical and numerical results have implications for the transmission of pulsed light through arrangements of nanoscale apertures with plasmonic coupling.

The conventional scintillation, or intensity fluctuation, that occurs in random electromagnetic beams is just one member of a broader class of four interconnected, polarization-resolved scintillations. We examine these generalized scintillations, called Stokes scintillations, that occur when two stochastic electromagnetic beams are made to interfere in Young’s experiment. We find that the magnitude of the conventional scintillation can be decreased, within certain limits, at the expense of an increase of one or more of the other Stokes scintillations. For certain applications however, it may be beneficial to suppress the latter.

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.

Highly nonparaxial electromagnetic fields are increasingly met in modern nanophotonics. Their characterization is vital, but the traditional beam-field methods are not applicable. In this work we d...

We propose a free-space, inverse design of nanostructure's effective mode-matching fields via a backward propagation of tightly focused vector beams to the pupil plane of an aplanatic system of high numerical aperture. First, we study the nanostructure's eigenmodes without considering any excitation fields and then extract the modal near fields in the focal plane. Each modal field is then taken as the desired focal field, the band-limited waves of which are backward propagated to the pupil plane via a reversal of the Richards--Wolf vector diffraction formula. The pupil fields can be designed to be genuinely paraxial by associating the longitudinal electric/magnetic field component with the radial one on the reference sphere. The inversely designed pupil field in turn is propagated forwardly into the focal region to generate the designed focal field, whose distribution over the nanostructure's surface is used to evaluate the overlap between the designed focal field and the modal fields, i.e., the modal expansion coefficients. Studies for a silicon nanodisk monomer, dimer, and tetramer demonstrate the ability of our inverse approach to design the necessary tightly focused vector field that can effectively and exclusively match a certain eigenmode of interest. Compared with the forward beam-shaping method, the inverse design approach tends to yield quantitatively more precise mode-matching field profiles. This work can have a significant impact on optical applications that rely on controllable and tunable mode excitation and light scattering., 12 pages, 9 figures

We consider two partially correlated scalar light beams in a spatially unitary interference setup. We introduce a state vector in a Poincaré-sphere-like geometrical configuration that fully specifies such an optical system and its evolution under spatial unitary transformations. We also identify three particular unitary operations together with their geometrical representations that can be optically implemented to realize an arbitrary spatial unitary transformation. Our work forms an advantageous geometrical platform to characterize distinguishability, visibility, degree of coherence, and classical entanglement, as well as their spatial unitary evolutions, in scalar two-beam light interference.

Wavelength selection has a major impact on spectral beam combining (SBC) of laser beams that propagate in free space and in the atmosphere. We derive analytical expressions for the intensity and four characteristic quality parameters of free-space SBC beams. We show that the peak intensity, the mean-square beam width, and the M 2 factor of an SBC beam at the target are all larger than those of one single laser beam of the same power and average wavelength. The peak intensity can be made higher by increasing the wavelength spacing but reducing the mean wavelength. However, when an SBC laser beam propagates in the atmosphere, the beam quality depends on the absorption coefficient of the multi-wavelength field due to thermal blooming. A thermal distortion number is introduced for an SBC beam and its validity is confirmed. We show that the power proportion of a high-absorption element within an SBC beam should be as low as possible to reduce the thermal blooming effect.

The groundbreaking research and ideas introduced by Emil Wolf continue to inspire researchers and motivate ongoing research in the wave properties of light. This special issue commemorates the legacy of Emil Wolf with research in physical optics, with specific focus on those areas where Wolf was active, such as optical coherence theory, inverse problems, singular optics, imaging, and polarization, and the intersection of these fields of study. Here we discuss the life of Emil Wolf and his influence on optical science and the optics community.

A simple phenomenological model, justified by rigorous electromagnetic analysis, is used to examine far-field polarization modulation introduced by a metallic screen with two subwavelength slits in uniform, partially polarized illumination. The accuracy of this intuitive model is assessed numerically, by employing the Fourier modal method, as regards the wavelength, the screen metal, and the slit depth, width, and separation. The Stokes parameters and the degree of polarization are evaluated paraxially in the far zone of the slits. We demonstrate that a uniform, partially linearly polarized incident beam can be rendered unpolarized by the nanoslits. Our work provides insight into the pivotal role that surface plasmons have in polarization modulation in photonics.

We investigate the two-point polarization correlation properties of random, statistically stationary, two-dimensional (2D) and three-dimensional (3D) electromagnetic fields. To this end two specific mutual polarization correlation functions are introduced, one based on a geometric approach using the Poincar\'e vectors and the other on the energy distribution employing the Jones vectors. Both approaches provide equivalent information about the space-time polarization correlations and enable one to associate with the field a polarization length over which the equal-time polarization states remain essentially unchanged. For fields obeying Gaussian statistics, the polarization correlation functions take on simple forms in terms of measurable quantities representing electromagnetic coherence. The formalism is demonstrated with two examples: blackbody radiation inside a cavity and in the far zone of a cavity wall aperture.

We propose a method for measuring the spatial coherence of light by means of temporal modulation of a double slit displayed on a digital micromirror device. It is demonstrated theoretically and experimentally that the technique is generally insensitive to background light, and thus its suppression or subtraction is not necessary. Moreover, the visibility of the interference fringe pattern can be enhanced by modulating only either one of the two slits. These favorable features enable one to measure the spatial coherence of even faint light more conveniently and accurately.

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.

We analyze the spin structure of weakly and tightly focused random light beams in view of the characteristic decomposition of the $3\ifmmode\times\else\texttimes\fi{}3$ polarization matrix. We take the incident beam to be a mixture of mutually uncorrelated, orthogonally linearly polarized parts which entails zero spin for the beam but leads to a nonzero purely transverse spin in focusing via the generation of a longitudinal electric-field component. We show that the spin ensuing from the so-called discriminating component of the polarization matrix displays a vortex structure for any degree of polarization and level of focusing, while the spin distribution of the pure component may vary from linear to circular as the incident beam changes from fully polarized to completely unpolarized. The spin of the discriminating component is related to the recently introduced concept of polarimetric nonregularity and emerges only when the degree of nonregularity is different from zero. Our results demonstrate the relationship between the rich polarimetric structure of random three-component light and the composition of the optical electric-field spin.

We examine the spin angular momentum (SAM) density associated with the recently introduced [Phys. Rev. A 100, 053833 (2019)], partially coherent surface-plasmon-polariton (SPP) vortex fields at a metal-air interface. We show that the vortices appearing in such structured SPP fields induce a SAM density both in the interface plane and in the direction normal to the interface. We find that the radial and azimuthal SAM densities are caused solely by the SPP electric-field correlations. However, besides the intrinsic spin component induced by the complex SPP wave vector, the azimuthal SAM density remarkably carries also a spin component created by the elementary SPPs comprising the partially coherent vortex field. The normal SAM density, on the other hand, arises mainly due to the SPP magnetic-field correlations. Our analysis specifically demonstrates that the state of coherence of the partially coherent SPP vortex field plays an essential role in shaping the SAM density distributions. Our findings can find applications to near-field particle manipulation and in spin-based integrated photonic circuit design.

Funding Information: The work was funded by the Academy of Finland (Projects 322002, 308393, 310511). The work is part of the Academy of Finland Flagship Programme, Photonics Research and Innovation (PREIN), decisions 320166 and 320165. Publisher Copyright: © 2021 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft. We study beaming properties of laser light produced by a plasmonic lattice overlaid with organic fluorescent molecules. The crossover from spontaneous emission regime to stimulated emission regime is observed in response to increasing pump fluence. This transition is accompanied by a strong reduction of beam divergence and emission linewidth due to increased degree of spatial and temporal coherence, respectively. The feedback for the lasing signal is shown to be mainly one-dimensional due to the dipolar nature of the surface lattice resonance. Consequently, the beaming properties along x and y directions are drastically different. From the measurements, we obtain the M 2 value along both principal directions of the square lattice as a function of the pump fluence. Our work provides the first detailed analysis of the beam quality in plasmonic lattice lasers and reveals the underlying physical origin of the observed strong polarization dependent asymmetry of the lasing signal.

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.

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.

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.

A method is proposed for spatial coherence measurement by means of temporal modulation of a double slit on a digital micromirror device. Some advantageous features including immunity to background light are shown theoretically and experimentally.

We introduce a phase-contrast ghost-imaging scheme for the characterization of temporal phase objects in terms of intensity correlations at two photodetectors. The technique is analogous to Zernike's phase-contrast imaging method and is based on utilizing a suitable filter function which renders the small-amplitude phase variations visible in the intensity correlation function. The approach is insensitive to temporal distortions and offers a promising method to analyze the phases of optical pulses.

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.

Polarimetric dimension and nonregularity are newly introduced concepts that characterize three-dimensional (3D) polarization states of light. We analyze the spectral polarimetric dimension and the degree of nonregularity associated with two kinds of tightly focused beams: A radially fully polarized Gaussian Schell-model (GSM) beam and a partially polarized beam composed of an incoherent superposition of two orthogonally polarized (coherent) plane-wave modes. We show that for both beams the focal field can exhibit genuine 3D and nonregular character, with even perfect nonregularity encountered for the tightly focused two-mode beam. These features originate from the partial spatial coherence and partial polarization of the incident beams, and in the limit of full coherence and polarization the three-dimensionality and nonregularity of the focal field vanish. We also find that the GSM beam can generate a nanoscale region around the focus where the field is essentially 3D unpolarized. The results demonstrate the rich polarimetric structure of focal fields and may find uses in optical particle manipulation and sensing.

We consider random, three-component, planar, quasihomogeneous electromagnetic sources and the nonparaxial far fields which they produce. We show that by measuring the far-field spectral polarization matrix on a spherical surface throughout a half-space enables to reconstruct the low spatial-frequency parts of the degrees of correlation between the parallel source field components. Further, detecting, in addition, the far-zone angular cross-spectral density matrix for all pairs of directions with significant intensity allows to deduce the low spatial-frequency contributions to the spectral densities of the source field components as well as the degrees of correlation between the orthogonal components. The validity of the formalism is demonstrated by considering a three-component, electromagnetic Gaussian Schell-model source. The results are important for the research on electromagnetic wide-angle radiators such as many thermal and fluorescent sources and light-emitting diodes.

We develop a scaling law for a class of statistically nonstationary scalar optical fields, which ensures spectral invariance on their propagation into the far zone of a planar source. The invariance involves the constraint that the normalized far-zone spectrum must be the same in every direction of observation, as well as equal to the normalized area-averaged source spectrum. Thus, it additionally represents an extension of the earlier work by Wolf on stationary fields [Phys. Rev. Lett. 56, 1370 (1986)PRLTAO0031-900710.1103/PhysRevLett.56.1370] that assumed the normalized source spectrum as independent of position. We present examples of both nonstationary and stationary fields that satisfy the scaling law and extended spectral invariance. acceptedVersion

We study the spectral and temporal coherence effects in the passage of a Gaussian Schell-model (GSM) scalar, plane-wave pulse train through a slab of nonlinear optical crystal exhibiting second-harmonic generation. We show that due to the nonlinear interaction the temporal and spectral degrees of coherence of the fundamental (F) and second-harmonic (SH) pulse trains at the exit facet may deviate markedly from the GSM and the global degree of coherence of the SH wave generally decreases with increasing incident F beam intensity. In addition, we find that due to the partial coherence of the incident GSM field the transmitted SH wave may show a double-peaked intensity distribution.

We present optical systems, which transform isodiffracting nonstationary beams into fields obeying either cross-spectral purity or spectral invariance. The designs are hybrid refractive-diffractive imaging systems, which are able to perform the desired transformations over a broad spectral bandwidth and irrespective of the state of spatial coherence of the input beam.

We study the field that is produced by a paraxial refractive axicon lens. The results from geometrical optics, scalar wave optics, and electromagnetic diffraction theory are compared. In particular, the axial intensity, the on-axis effective wavelength, the transverse intensity, and the far-zone field are examined. A rigorous electromagnetic diffraction analysis shows that the state of polarization of the incident beam strongly affects the transverse intensity distribution, but not the intensity distribution in the far zone. (C) 2017 Optical Society of America

The subject of this review lies at the cross-roads of optical coherence theory and surface wave physics. We first recall how electromagnetic surface waves affect the spectrum, polarization, and spatial correlation properties of thermal near fields. We then discuss the modulation, control, and measurement of spatial coherence of random optical beams by surface plasmon polaritons (SPPs). We review the spectral polarization and subwavelength coherence of three-dimensional evanescent fields. Finally, we examine polychromatic, structured SPP fields of tailored electromagnetic coherence.

The spectral and characteristic decompositions of the polarization matrix provide fruitful frameworks for the physical interpretation of three-dimensional (3D) partially polarized light fields. The decompositions are formulated in terms of the three pure eigenstates, which in turn are represented through their associated orthogonal complex 3D Jones vectors. This mathematical orthogonality does not correspond, in general, to orthogonality of the polarization-ellipse planes of the respective eigenstates. Consequently, due to such inherent mathematical complexity, the geometric and physical interpretation of these sets of orthogonal complex vectors, being essential for the best understanding of the structure and properties of partially polarized 3D light, has not been addressed thoroughly. In this work, the geometric and physical features of sets of three orthonormal 3D Jones vectors are identified and analyzed, allowing one to obtain meaningful interpretations of any given mixed (partially polarized) 3D polarization state in terms of either the spectral or the characteristic decompositions. Among other results, it is found that, given a pure polarization state, any plane in space contains the polarization ellipse of a pure state that is orthogonal to it, and the mathematical expressions for the azimuth and ellipticity of such an ellipse are calculated in terms of the angular parameters determining said plane and the ellipticity of the given state. Furthermore, the spin vectors of the three polarization eigenstates are arranged in a peculiar spatial manner, such that they lie in a common plane. Beyond polarization phenomena, the approach presented also has potential applications in areas where 3 \ifmmode\times\else\texttimes\fi{} 3 unitary matrices play a key role, like three-level quantum systems and gates for ternary quantum logic circuits.

We introduce a class of partially coherent surface plasmon polariton (SPP) fields carrying optical vortices, generated through a judicious superposition of planar SPPs with a prescribed initial phase distribution and arbitrary correlations at a metal-air interface. We explore the global degree of coherence, energy densities, energy flows, orbital and spin angular momenta, and the polarization states of such partially coherent SPP vortex (SPPV) fields in terms of their coherent mode representation. The salient physical characteristics of these SPPV fields can be widely tuned by controlling the constituent SPP correlations, tailoring such SPPV fields to desired applications in statistical nanoplasmonics.