Theory of partially coherent electromagnetic fields in the space-frequency domain

Synthesis of electromagnetic Schell-model sources

A procedure for the synthesis of the most general electromagnetic Schell-model light source is proposed. It makes use of the generalized van Cittert-Zernike theorem to produce the electromagnetic source starting from a primary spatially incoherent source, characterized by a suitable position-dependent polarization matrix. By resorting to the spectral decomposition of the polarization matrix, it is shown how such an incoherent source can be synthesized by using a Mach-Zehnder interferometer, with suitable amplitude transmittances placed in its arms, fed by two mutually uncorrelated laser beams. Examples are given for the case of electromagnetic Gaussian Schell-model sources.

Depolarization of quasi-monochromatic light by thin resonant gratings

Using rigorous diffraction theory, we show that it is possible to depolarize quasi-monochromatic light using a fully deterministic, thin optical element. The principle is based on polarization-sensitive resonant gratings in total-internal-reflection configuration, and careful numerical design by the Fourier modal method. We present numerical examples for 0.1 nm and 10 nm bandwidths.

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.

Computationally efficient coherent-mode representations

The numerical calculation of traditional coherent-mode representations (CMRs) involves an eigenvalue decomposition of the cross-spectral density matrix. An efficient alternative modal representation of a partially coherent field can be realized using an LDL decomposition. Storage requirements are reduced by an amount on the order of the ratio between the coherence length and the source width. The efficiency of calculations requiring a CMR (e.g., numerical evaluation of partially coherent propagation effects) may thus be significantly improved, particularly when low-coherence fields are considered.

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.

The van Cittert-Zernike theorem for electromagnetic fields

The van Cittert-Zernike theorem, well known for the scalar optical fields, is generalized for the case of vector electromagnetic fields. The deduced theorem shows that the degree of coherence of the electromagnetic field produced by the completely incoherent vector source increases on propagation whereas the degree of polarization remains unchanged. The possible application of the deduced theorem is illustrated by an example of optical simulation of partially coherent and partially polarized secondary source with the controlled statistical properties.

Partially correlated azimuthal vortex illumination: coherence and correlation measurements and effects in imaging

Correlations in the illumination field have a profound impact on the image contrast for features near the resolution limit. The pupil polarization affects these correlations. We show that a polarization vortex has a particularly dramatic effect. A theoretical model is given for the correlation matrix of a partially correlated source created by placing an azimuthal polarization vortex mode converter in the pupil plane of a critical illumination system. We then validate this model experimentally using a reversed-wavefront Young interferometer, directly show the impact that the phase of the correlation function has on image contrast.

Partially correlated sources with complete polarization

We show that the correlation matrix of any electromagnetic planar source endowed with complete polarization must be of a universal form, no matter how the field components are correlated across the source plane. Such form entails that both the realizability conditions and the search of vector modes of the source reduce to scalar problems. We shall also briefly dwell on how propagation affects the polarization properties of the field and, in particular, we discuss cases in which propagated light remains completely polarized.

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.

Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems

The change of coherence and polarization of an electromagnetic beam modulated by a random anisotropic phase screen passing through any optical system is found within the framework of complex ABCD-matrix theory This means that the formalism can treat imaging and Fourier transform and free-space optical systems, as well as fractional Fourier transform systems, with finite-size limiting apertures of Gaussian transmission shape. Thus, the current paper shall be considered as a continuation, extension, and generalization of a previous work by Shirai and Wolf [J. Opt. Soc. Am. A21, 1907 (2004)]. It will be shown that the inclusion of apertures in the optical system strongly influences not only the propagation of spatial coherence but also the degree of polarization of a propagating field. Analytical expressions of coherence and polarization propagation will be given in terms of the matrix elements for any complex optical system.

Modal expansion for J0-correlated electromagnetic sources

We introduce a class of sources that turn out to be J0 correlated in a scalar description but exhibit varied correlation properties when examined in electromagnetic terms. We prove that a modal expansion can be built explicitly for any of these sources and give an example to illustrate the richness of their behavior.

Can a light beam be considered to be the sum of a completely polarized and a completely unpolarized beam?

In a classic paper that may be regarded as the starting point of polarization optics, G. G. Stokes [Trans. Cambridge Philos. Soc.9, 399 (1852)] presented a theorem according to which any light beam is equivalent to the sum of two light beams, one of which is completely polarized and the other completely unpolarized. We show that Stokes' proof of this theorem is flawed. We present a condition for the theorem to be valid.

Communication modes with partially coherent fields

We develop a theory for the description of partially coherent wave fields in linear optical systems in terms of the so-called communication modes. The communication modes are the singular functions and singular values of the appropriate propagation kernels. In particular, we show that optical fields of any state of coherence may be readily propagated through deterministic systems using the modal representation based on the system properties. The relation of the communication modes to the conventional coherent-mode representation is discussed, and expressions for the effective degree of coherence in the optical system are derived. The results are illustrated by numerical examples in optical near-field geometry.

Overall degree of coherence for vectorial electromagnetic fields and the Wigner function

We elaborate the overall degree of coherence for vectorial electromagnetic waves within paraxial approximation, expressing it in terms of a polarization Wigner function and the spatial-angular Stokes parameters.

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.

Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices

We assess the degree of coherence of vectorial electromagnetic fields in the space-frequency domain as the distance between the cross-spectral density matrix and the identity matrix representing completely incoherent light. This definition is compared with previous approaches. It is shown that this distance provides an upper bound for the degree of coherence and visibility for any pair of scalar waves obtained by linear combinations of the original fields. This same approach emerges when applying a previous definition of global coherence to a Young interferometer.

Maximizing Young's fringe visibility through reversible optical transformations

When a Young's interferometer is fed by an electromagnetic beam, fringes of low, or even zero, visibility do not necessarily indicate lack of correlation between two typical field components at the pinholes. The passage of light that emerges from one of the pinholes through a nonabsorbing anisotropic optical element may enhance the visibility. We inquire about the maximum visibility that can be attained through such a reversible transformation (polarizers being excluded). We find that such a quantity can be evaluated in closed form. Its value is directly related to the Ky Fan 1-norm of the correlation matrix of the illuminating beam.

Propagation theories of arbitrary-order correlations of partially coherent electromagnetic fields based on separated-coordinate mode decomposition

Theories to calculate the propagation of arbitrary-order correlations of stationary or nonstationary partially coherent electromagnetic fields are proposed. The theories are based on separated-coordinate mode decomposition, and can make the well-developed propagation theories of fully coherent electromagnetic fields applicable to partially coherent electromagnetic fields governed by linear Maxwell equations. The validity of the theories is illustrated by an example.

Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations

We propose to analyze Shannon entropy properties of partially coherent and partially polarized light with Gaussian probability distributions. It is shown that the Shannon entropy is a sum of simple functions of the intensity, of the degrees of polarization, and of the intrinsic degrees of coherence that have been recently introduced. This analysis clearly demonstrates the contribution of partial polarization and of partial coherence to the characterization of disorder of the light provided by the Shannon entropy, which is a standard measure of randomness. We illustrate these results on two simple examples.

Contrasts of Stokes parameters in Young's interference experiment and electromagnetic degree of coherence

We analyze the modulation of the Stokes parameters in Young's two-pinhole interference experiment with a random electromagnetic beam. We demonstrate that the electromagnetic (spectral) degree of coherence put forward in Opt. Lett.29, 328 (2004) [or its space-time analog in Opt. Express 11, 1137 (2003)] is physically related to the contrasts of modulation in the four Stokes parameters. More explicitly, the electromagnetic degree of coherence is a measure of both the visibility of the intensity fringes and the modulation contrasts of the three polarization Stokes parameters. We also show that by using suitable wave plates the modulation in any Stokes parameter can be transformed into the form of intensity variation, and hence the electromagnetic degree of coherence can be obtained experimentally by four visibility measurements.

Propagation theories of partially coherent electromagnetic fields based on coherent or separated-coordinate mode decomposition

Propagation theories of partially coherent electromagnetic fields based on coherent mode decomposition or separated-coordinate mode decomposition are proposed. With the proposed propagation theories, various powerful theories for the propagation of fully coherent electromagnetic fields can be used for the propagation of partially coherent electromagnetic fields. The proposed theories are applicable to any propagation problem of partially coherent electromagnetic fields governed by linear Maxwell equations. Some examples are provided to illustrate the validity of the proposed theories.

Spatial coherence effects in light scattering from metallic nanocylinders

We study the scattering of a partially coherent electromagnetic beam from metallic nanocylinders and analyze the effects of plasmon resonances on the coherence and polarization properties of the optical near field. We employ the coherent-mode representation for the incident field and solve the scattering problem independently for each mode by using a boundary-integral method. Our results show that the plasmon resonances may significantly affect the coherence and polarization characteristics of the near field and that partial coherence influences the energy flow in nanocylinder arrays.

Vector mode analysis of a young interferometer

It is proved that, when the vector modal theory of coherence is applied to a pair of fixed points, exact results are obtained for the mode structure. In particular, it is shown that the field radiated by the pinholes of a Young interferometer can always be represented by the incoherent superposition of no more than four perfectly correlated and polarized modes. The role of such modes is illustrated through a simple example.

Effects of coherence on the degree of polarization in a young interference pattern

Recent predictions concerning the relationship between the degree of polarization at a typical point of a Young interference pattern and the degree of coherence of the electromagnetic field at the pinholes are tested by a simple experiment. In particular, it is shown that light that is completely unpolarized at the pinholes can become partially polarized across the fringe pattern.

Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations

The spatiotemporal coherence properties of partially polarized light with Gaussian probability distributions are analyzed using the mutual information that is a standard measure of statistical dependence. This approach leads to intrinsic degrees of coherence that have powerful invariance properties and that provide new information in comparison with other recently introduced degrees of coherence.

Connection between electric and magnetic coherence in free electromagnetic fields

We introduce quantitative measures for the description of the electric and magnetic coherence in a stationary, random electromagnetic field at two points, in a volume, and in the Fourier space. These quantities are applied to free electromagnetic fields, and several theorems regarding the relationship between the two types of coherences in such fields are established. Fields which are statistically homogeneous, and those which, in addition, are statistically isotropic are considered separately. Furthermore, the connection between the electric and magnetic coherence is exemplified for some specific statistically homogeneous fields.

Theory of spatially and spectrally partially coherent pulses

A coherent-mode representation for spatially and spectrally partially coherent pulses is derived both in the space-frequency domain and in the space-time domain. It is shown that both the cross-spectral density and the mutual coherence function of partially coherent pulses can be expressed as a sum of spatially and spectrally and temporally completely coherent modes. The concept of the effective degree of coherence for nonstationary fields is introduced. As an application of the theory, the propagation of Gaussian Schell-model pulsed beams in the space-frequency domain is considered and their coherent-mode representation is presented.

Coherent-mode representation of a statistically homogeneous and isotropic electromagnetic field in spherical volume

It is known that statistically stationary, homogeneous, and isotropic source distributions generate, in an unbounded low-loss medium, an electromagnetic field whose electric cross-spectral density tensor is proportional to the imaginary part of the infinite-space Green tensor. Using the recently established electromagnetic theory of coherent modes, we construct, in a finite spherical volume, the coherent-mode representation of the random electromagnetic field having this property. The analysis covers the fundamental case of blackbody radiation but is valid more generally; since a thermal equilibrium condition is not invoked, the electromagnetic field may have any spectral distribution. Within the scalar theory of coherent modes, which has been available for more than two decades, the analogous formulation results in the first explicit three-dimensional coherent-mode representation.

Electromagnetic coherence theory of laser resonator modes

A theory of open laser resonators is formulated within the framework of the electromagnetic coherence theory. It is shown that if only one Fox-Li mode contributes to the field at a given frequency, then the field at that frequency is necessarily completely coherent in view of the space-frequency counterpart of the recently introduced degree of coherence of electromagnetic fields [Opt. Express 11, 1137 (2003)]. It is also shown that the relation between the number of Fox-Li modes and the new degree of coherence is analogous to the relation established in the scalar theory of laser resonator modes. Difficulties that arise with the formerly introduced visibility-based definition of the electromagnetic degree of coherence are briefly discussed.