Maximizing Young's fringe visibility through reversible optical transformations

Double-slit interference experimental platform based on a Sagnac interferometer

In this paper, we build a Young's double-slit experimental platform with adjustable multiparameters by inserting a single slit in a Sagnac interferometer. Our experiment proves that this platform can easily control the parameters related to double slits. In addition, a new, to the best of our knowledge, type of double-slit experiment is performed on the platform. Our approach provides a detailed conceptual and experimental analysis for wave-particle duality and will be useful for research on quantum optics.

Super cosh-Gauss nonuniformly correlated radially polarized beam and its propagation characteristics

In this paper, a new kind of partially coherent vector beam termed as super cosh-Gauss nonuniformly correlated radially polarized (SCNRP) beam is introduced. Such beam source exhibits almost perfect coherence between two points that are within the beam center region or located on a ring concentric with the beam center. However, the coherence drops or even vanishes when the two points leave the central region and are located on the concentric rings with different radii. The second-order statistical properties, such as the spectral density, the state of polarization (SOP), and the degree of polarization (DOP) of such beam upon free-space propagation are studied through numerical examples. Our results reveal that the beam displays a self-focusing property during propagation. The focusing ability can be enhanced with increasing the beam index and decreasing the beam's spatial coherence width, whereas the DOP and SOP remain unchanged on propagation. Meanwhile, we establish an experimental system with the use of a radial polarization converter and a digital micro-mirror device to synthesize the SCNRP beam with controllable beam index and spatial coherence width. The spectral density and polarization properties of the synthesized beam during propagation are measured and analyzed in the experiment. The experimental results agree well with our theoretical predictions.

Statistical properties of a partially coherent radially polarized vortex beam propagating in a uniaxial crystal

Free-space propagation and experimental generation of a partially coherent radially polarized (PCRP) vortex beam were studied recently [Opt. Express24, 13714 (2016)OPEXFF1094-408710.1364/OE.24.013714]. In this work, we explore the statistical properties of such a PCRP vortex beam propagating in a uniaxial crystal. We show that the anisotropy of the refractive index of the uniaxial crystal induces the asymmetrical distribution of the intensity, the degree and the state of polarization, as well as the degree of coherence of the beam during propagation. Further, by comparing the asymmetrical distribution of the statistical properties of the PRCP vortex beam with those of a PRCP beam without a vortex phase, we find that the asymmetrical features can be used for determining whether a PCRP beam carries the vortex phase. Further, we show that from the far-field distribution of the degree of coherence, we could quantify the topological charge and distinguish the handedness of the vortex phase. Our findings provide a novel approach for measuring the phase information of the partially coherent vortex beams.

Maximizing Young fringe visibility with a universal SU2 polarization gadget

When a Young interferometer is fed by a general electromagnetic field, the fringe visibility may change upon insertion of an anisotropic optical element over one of the interferometer pinholes. The maximum visibility that the fringes may exhibit in this way is theoretically known, but no direct experimental check seems to be available. Here we discuss the scheme of an experimental test. In particular, maximum fringe visibility is obtained with a three-component universal SU2 polarization gadget for arbitrary coherence features of the illuminating field. Confirming results obtained with a suitable experimental setup are presented.

Measuring vector field correlations using diffraction

We present a method for efficiently measuring the 2 × 2 correlation matrix for paraxial partially coherent beams by using diffraction from small apertures and obstacles. Several representations for this matrix function of four spatial variables are discussed and illustrated with experimental results, including various alternative definitions of the spatial degree of coherence.

Resource-theoretic approach to vectorial coherence

We propose a formal resource-theoretic approach to assess the coherence between partially polarized electromagnetic fields. From this framework, we identify two resource theories for the vectorial coherence: polarization-sensitive coherence and polarization-insensitive coherence. For each theory, we find the set of incoherent states and a class of operations that preserve this set (i.e., the incoherent operations). Both resource theories are endowed with a certain preorder relation that provides a hierarchy among the coherence-polarization states; thus, a necessary condition to consider in deciding whether a quantity is proper to measure the vectorial coherence is that it respects such a hierarchy. Finally, we examine most previously introduced coherence measures from this perspective.

Evolution properties of polarization states of far-zone electromagnetic field scattered from an anisotropic medium

Within the accuracy of the first-order Born approximation, expressions are derived for the cross-spectral density matrix of the electromagnetic field scattered from a three-dimensional (3D), spatially anisotropic medium. By using the analytical forms of the spectral degree of polarization (SDOP), semi-major axis, semi-minor axis, and orientation angle of the polarization ellipse, we illustrate the evolution properties of the polarization states of the far-zone scattered field. Dependences of the polarization ellipses on different combinations of correlation lengths of the medium and polarization of incident plane wave are also revealed through numerical simulations. It is shown that the polarization ellipse of the scattered field can be either stretched or squeezed along certain scattering angles, strongly depending on the anisotropic statistics of medium and polarization of incident plane wave. Our results substantially enrich the study on the reciprocal relations between the far-zone scattered properties and correlation statistics of a spatially anisotropic medium.

What is the maximum attainable visibility by a partially coherent electromagnetic field in Young's double-slit interference?

What is the maximum visibility attainable in double-slit interference by an electromagnetic field if arbitrary - but reversible - polarization and spatial transformations are applied? Previous attempts at answering this question for electromagnetic fields have emphasized maximizing the visibility under local polarization transformations. I provide a definitive answer in the general setting of partially coherent electromagnetic fields. An analytical formula is derived proving that the maximum visibility is determined by only the two smallest eigenvalues of the 4×4 two-point coherency matrix associated with the electromagnetic field. This answer reveals, for example, that any two points in a spatially incoherent scalar field can always achieve full interference visibility by applying an appropriate reversible transformation spanning both the polarization and spatial degrees of freedom - without loss of energy. Surprisingly, almost all current measures predict zero-visibility for such fields. This counter-intuitive result exploits the higher dimensionality of the Hilbert space associated with vector - rather than scalar - fields to enable coherency conversion between the field's degrees of freedom.

Electromagnetic theory of optical coherence [Invited]

The coherence theory of random, vector-valued optical fields has been of great research interest in recent years. In this work we formulate the foundations of electromagnetic coherence theory both in the space-time and space-frequency domains, with particular emphasis on various types of optical interferometry. Analyzing statistically stationary, two-component (paraxial) electric fields in the classical and quantum-optical contexts we show fundamental connections between the conventional (polarization) Stokes parameters and the associated two-point (coherence) Stokes parameters. Measurement of the coherence and polarization properties of random vector beams by nanoparticle scattering and two-photon absorption is also addressed.

Linear algebraic theory of partial coherence: continuous fields and measures of partial coherence

This work presents a linear algebraic theory of partial coherence for optical fields of continuous variables. This approach facilitates use of linear algebraic techniques and makes it possible to precisely define the concepts of incoherence and coherence in a mathematical way. We have proposed five scalar measures for the degree of partial coherence. These measures are zero for incoherent fields, unity for fully coherent fields, and between zero and one for partially coherent fields.

Effective degree of coherence: a second look

In this paper, we show that the most general set of transformations of electromagnetic fields, for which overall (global) second-order coherence properties can reasonably be expected to remain unchanged, is the set of scaled unitary transformations. Building on our earlier results concerning coherence functionals that are invariant to scaled unitary transformations, we prove that the effective degree of coherence is the only such functional that is "additive" in the sense that it can be computed for linear combinations of fields from its values for pairwise sums of the constituent fields. Additionally, we highlight the fact that the invariance of the effective degree of coherence to scaled unitary transformations means that it has the same value when computed from most of the important representations of electromagnetic fields. We then go on to use the effective degree of coherence to provide a generalization of the scalar two-point degree-of-coherence function to a system consisting of two orthogonal Hilbert spaces. Interestingly, several commonly used measures of coherence and polarization turn out to be special cases of this generalization.

Isotropic bicomponent partially polarized and partially coherent light

The degree of coherence of scalar light remains constant when the fields are modified by the same random linear transformation, which can be represented by the multiplication by a random complex number. This shows that the coherence properties of scalar light at order two are not modified with the increase of disorder of each field that results from these transformations. We analyze the generalization of this property to partially polarized light. We determine the class of fields that can possess this property for any couple of points in a space-frequency or space-time domain after modification with deterministic Jones transformations. We show that the second-order coherence properties of this class of light can be generated experimentally with two uncorrelated totally polarized sources that have the same scalar coherence properties.

Coherence-polarization mixing in resonance gratings

We show, using rigorous diffraction theory, that resonance gratings can be used to transfer partial spatial correlation to partial polarization even if the incident light beam is fully polarized. The phenomenon is based on the fact that either of the two orthogonal polarization components can be coupled into the leaky waveguide mode, leading to a strong phase delay, while the other one is reflected without being coupled into the grating. Numerical demonstrations are based on a Gaussian Schell-model beam and a grating analysis performed by rigorous Fourier modal method.

Coherence and visibility for vectorial light

Two-path interference of transversal vectorial waves is embedded within a larger scheme: this is four-path interference between four scalar waves. This comprises previous approaches to coherence between vectorial waves and restores the equivalence between correlation-based coherence and visibility.

A complete characterization of pre-Mueller and Mueller matrices in polarization optics

The Mueller-Stokes formalism that governs conventional polarization optics is formulated for plane waves, and thus the only qualification one could require of a 4 x 4 real matrix M in order that it qualify to be the Mueller matrix of some physical system would be that M map Omega((pol)), the positive solid light cone of Stokes vectors, into itself. In view of growing current interest in the characterization of partially coherent partially polarized electromagnetic beams, there is a need to extend this formalism to such beams wherein the polarization and spatial dependence are generically inseparably intertwined. This inseparability brings in additional constraints that a pre-Mueller matrix M mapping Omega((pol)) into itself needs to meet in order to be an acceptable physical Mueller matrix. These additional constraints are motivated and fully characterized.

Elementary-field expansions of genuine cross-spectral density matrices

A necessary and sufficient nonnegative definiteness condition for the cross-spectral density matrix (CDM) is derived. It is also shown that this realizability condition allows the expansion of genuine CDMs in terms of recently introduced elementary fields, namely, mean-square coherent beams, and fields with position-independent stochastic behavior. The special case of uniformly polarized electromagnetic Schell-model sources is also analyzed.

Correlation matrices of completely unpolarized beams

The simplest form for the correlation matrix of a completely unpolarized electromagnetic beam is the product of a scalar correlation function times a unit matrix. We show, however, that classes of unpolarized beams exist for which the diagonal elements of the correlation matrix are not equal to each other and the off-diagonal elements do not vanish identically. This gives rise to a distinction between pure and impure unpolarized beams. The two types of beams can be distinguished at the experimental level by their behavior in a Young's interferometer.

Modulation of coherence and polarization using liquid crystal spatial light modulators

We propose a method for modulation of coherence and polarization of electromagnetic fields, employing two crossed zero-twisted nematic liquid crystal spatial light modulators. In contrast to a similar technique analyzed by Shirai and Wolf [J. Opt. Soc. Am A, 21, 1907, (2004)] our method provides a wide range simultaneous modulation of coherence and polarization. The dependence of the obtained results on different definitions of electromagnetic coherence is considered.

Maximizing Young's fringe visibility under unitary transformations for mean-square coherent light

Given the values of the degree of polarization of the fields at the pinholes in a Young interferometer, the maximum attainable visibility under unitary transformations is determined when the illuminating beam is mean-square light. Analytical expressions are also obtained for both the field vector (in the mean-square sense) and the cross-spectral density matrix associated with this kind of beams. A comparative summary is also provided of the main characteristics of well-known types of random electromagnetic fields frequently handled in the literature.

Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields

We demonstrate monotonic, irreversible effects on the entropy, the degree of polarization, and different degrees of coherence when random, energy-preserving unitary transformations are applied to vectorial electromagnetic fields.

Equivalence between optimum Young's fringe visibility and position-independent stochastic behavior of electromagnetic fields

Stochastic electromagnetic fields characterized by optimized fringe visibility in a Young interferometric arrangement are shown to be those whose random character is position independent. The optimization procedure involves local unitary transformations, which can be implemented by using reversible anisotropic polarization devices placed at the two pinholes. It is also shown that the local degree of polarization in the optimized interferometer is constant across the superposition region and coincides with the degree of polarization at the two pinholes.

Mean-square coherent light

Mean-square coherent light is defined as light that is able to interfere with fringes of unit visibility when its electromagnetic field is multiplied by appropriate nonsingular deterministic Jones matrices. It includes light that satisfies the factorization condition at order one and partially polarized light that leads to interference fringes of unit visibility. A necessary and sufficient experimentally measurable condition to determine if two electromagnetic fields are mean-square coherent is established. Furthermore, different properties of partially polarized mean-square coherent light are discussed, such as its relation to the factorization condition and its evolution with propagation.

Irreversible effects of random modulation on coherence properties of partially polarized light

The effect of the multiplication of two vector electric fields by random Jones matrices on their coherence properties is investigated by analyzing their interference fringes. It is shown that these random modulations cannot increase the maximal visibility obtained when the polarization states of the two analyzed electric fields are optimized. This new result generalizes a standard property of perfectly polarized light to partially polarized light and is representative of the irreversible effects of random modulations.

Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields

In terms of the spectral density tensors associated with the electric field vector, the maximum visibility one can obtain in a two-point interference arrangement by using local (i.e., position-dependent) unitary transformations applied at such points is determined. It is also shown that the maximum visibility can be expressed in terms of a number of well-known parameters describing the coherence and polarization features of the field.

Relation between degrees of coherence for electromagnetic fields

In terms of the electric spectral density tensors associated with a random electromagnetic field, it is shown that, under certain conditions, a link can be established between two definitions of the degree of coherence for electromagnetic beams introduced not long ago by Wolf [Phys. Lett. A312, 263 (2003)] and by Setälä [Opt. Lett.29, 328 (2004)]. This connection is formulated as a certain equivalence relationship concerning the behavior of the above degrees of coherence and the degree of polarization of the field at two points.