Modified hollow Gaussian beam and its paraxial propagation

A model named modified hollow Gaussian beam (HGB) is proposed to describe a dark hollow beam with adjustable beam spot size, central dark size and darkness factor. In this modified model, both the beam spot size and the central dark size will be convergent to finite constants as the beam order approaches infinity, which are much different from that of the previous unmodified model, where the beam spot size and the central dark size will not be convergent as the beam order approaches infinity. The dependences of the propagation factor of modified and unmodified HGBs on the beam order are found to be the same. Based on the Collins integral, analytical formulas for the modified HGB propagating through aligned and misaligned optical system are derived. Some numerical examples are given.     

Parametric study of uniformly polarized stochastic electromagnetic beam and its imaging

A parametric study is performed in investigating the stochastic electromagnetic beam generated by a uniformly polarized electromagnetic Gaussian Schell-model source and passing through ABCD optical systems. Through theoretical analysis, the requirement is derived that the uniformly polarized electromagnetic field can be obtained at the output plane of the imaging optical system. Furthermore, the general imaging formula of the stochastic electromagnetic beam is derived. Numerical examples are also presented to illustrate the application.     

Partially coherent anomalous hollow beam and its paraxial propagation

Anomalous hollow beam is extended to the partially coherent case. Analytical propagation formulae for a partially coherent anomalous hollow beam passing through a paraxial ABCD optical system are derived. The propagation properties of a partially coherent anomalous hollow beam in free space and the focusing properties of a partially coherent anomalous hollow beam are studied numerically. It is found that the propagation and focusing properties of the partially coherent anomalous hollow beam are closely related to its initial coherence.     

Analytical structure of an apertured vector Gaussian beam in the far field

Based on the angular spectrum representation of the Maxwell’s equations and the complex Gaussian expansion of the aperture function, the structure of an apertured vector Gaussian beam in the far field is presented in the integral form. By means of the method of stationary phase, the analytical vectorial structures are obtained. According to the analytical expressions, the characteristics of vectorial structure of an apertured Gaussian beam are investigated in the far field. The influence of a linearly polarized angle on the vectorial structure is also studied in the far field. This research provides a novel approach to further comprehend the vectorial property of an apertured Gaussian beam.     

Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams

Several global parameters are proposed to characterize the radial and azimuthal polarization content of non-uniformly totally polarized beams. Such figures of merit can be written and measured in terms of two Stokes parameters, and also from the data at the output of either a radial or an azimuthal dichroic polarizer, integrated throughout the beam profile. The measurability of the proposed parameters has also been experimentally checked.     

Generalized beam propagation factors of truncated partially coherent cosine-Gaussian and cosh-Gaussian beams

Based on the method of truncated second-order moments, the analytical expressions of the generalized beam propagation factor for truncated partially coherent cosine-Gaussian and cosh-Gaussian beams are derived. The beam propagation factors of the truncated partially coherent cosine-Gaussian and cosh-Gaussian beams depend on the acentric parameter, the truncation parameter, and the coherent parameter. According to the derived expressions, the beam propagation factors are illustrated and analyzed with numerical examples, and the influences of the different parameters on the beam propagation factors are also discussed in detail.     

Statistical properties of correlated radial stochastic electromagnetic array beams on propagation

A kind of array beam named the correlated radial stochastic electromagnetic array beam that is generated by an electromagnetic Gaussian Schell-model source is introduced by use of tensor method. The analytical expression for the cross-spectral density matrix of this array beam propagating through the turbulent atmosphere and in free space is obtained after performing vector integration. Some typical numerical calculations are illustrated for the changes in the spectral density, spectral degree of polarization, and spectral degree of coherence of the beam on propagation. We find that the atmospheric turbulence can destroy the correlated effect among the beamlets.     

Overall parameters for the characterization of non-uniformly totally polarized beams

Several overall parameters are introduced to characterize the linear or circular polarization content of a non-uniformly totally polarized beam over the region of its wavefront where the irradiance is significant. These figures of merit are determined from the values of the Stokes parameters. The physical meaning of the proposed parameters is tested by computing some numerical examples, and their measurability is checked by considering non-uniformly totally polarized fields generated after propagation through uniaxial anisotropic materials.     

Vectorial structures of non-paraxial linearly polarized Gaussian beam and their beam propagation factors

Based on the vectorial structure of non-paraxial electromagnetic beam and non-paraxial vectorial moment theory, the relationship of the beam waists, the divergence angles and the beam propagation factors among non-paraxial linearly polarized Gaussian beam, its TE and TM terms have been presented, respectively. The analytical beam propagation factors are given and further discussed at the highly non-paraxial case. The maximum divergence angles in the x-direction of non-paraxial linearly polarized Gaussian beam, its TE and TM terms are all 54.7°, and those in the y-direction are limited to be 63.4°, 67.7° and 39.2°, respectively. As TE and TM terms are orthogonal and can be detached at the far field, the potential applications of the isolated TE and TM terms are deserved further investigation.     

Changes in generalized Stokes parameters of stochastic electromagnetic beams on propagation through ABCD optical systems and in the turbulent atmosphere

Using the derived formulas for the transformation of 2 × 2 cross-spectral density matrix of the stochastic electromagnetic beams propagating through ABCD optical systems and in the turbulent atmosphere, the changes in the generalized Stokes parameters of the beams propagating under these conditions can be investigated directly. Some typical numerical calculations are illustrated relating to the electromagnetic Gaussian Schell-model beams passing through free space, focal system, dual-focus system, and the turbulent atmosphere with different structure parameters. Further extensions are also pointed out.     

Off-Axis Astigmatic Gaussian Beam Combination Beyond the Paraxial Approximation

Taking the off-axis astigmatic Gaussian beam combination as an example, the beam-combination concept is extended to the nonparaxial regime. The closed-form propagation expressions for coherent and incoherent combinations of nonparaxial off-axis astigmatic Gaussian beams with rectangular geometry are derived and illustrated with numerical examples. It is shown that the intensity distributions of the resulting beam depend on the combination scheme and beam parameters in general, and in the paraxial approximation （i.e., for the small f-parameter） our results reduce to the paraxial ones.     

Electromagnetic fields that remain totally polarized under propagation

It is remarked that, in general, a totally polarized field becomes partially polarized at the output of an optical system, even though the field propagates through non-polarizing devices. Two kinds of fields are shown in the present work, which maintain the totally-polarized character at the output of any (deterministic) non-polarizing first-order optical system. This property is satisfied by uniformly totally polarized beams and also by those fields whose electromagnetic degree of coherence [T. Setälä, J. Tervo, A.T. Friberg, Opt. Lett. 29 (2004) 328.] equals 1.     

Effect of cross-polarization of electromagnetic source on the degree of polarization of generated beam

In this paper we study the degree of cross-polarization of a beam-like field whose cross-spectral density matrix is symmetric with respect to the permutation of its spatial arguments. Formulas were derived expressing the degree of cross-polarization in terms of the generalized Stokes parameters. With the help of an uniformly polarized quasi-homogeneous model source, the effect of degree of cross-polarization of a source on the degree of polarization of the radiated beam was demonstrated, that, two sources with same spectral degrees of coherence and polarization but with different degrees of cross-polarization can generate beams that have different spectral degrees of polarization in the far field.     

Spatial coherence properties of monochromatic electromagnetic beams and of laser modes

It is shown that, contrary to common belief, monochromatic light beams are, in general, not spatially completely coherent, i.e., they will, in general, not produce fringes of unit visibility in a Young's double pinhole interference experiment. We cite experiments with laser modes which confirm this result.     

The beam propagation factors and the kurtosis parameters of a Lorentz beam

Based on the second-order moments, the beam propagation factors and the kurtosis parameters of a Lorentz beam have been investigated. The M² value of a Lorentz beam is verified to be . The analytical expressions of the kurtosis parameters have been derived. The kurtosis parameter varies upon the propagation, and it is decided by two ratios z/z_rx and z/z_ry. The kurtosis parameter versus the two ratios z/z_rx and z/z_ry is plotted and discussed in detail. This work is of benefit to the practical application of laser sources that produce highly divergent beams.     

Tensor theory of electromagnetic radiometry

A theory of electromagnetic radiometry is built on the premise that the electromagnetic generalised radiance has a tensor structure, represented by the electric, magnetic and mixed generalised radiance tensors as fundamental quantities. They allow overcoming the limitations due to the scalar generalised radiances, proposed for characterizing stationary random electromagnetic sources. Furthermore, they provide a unified framework for completely describing the energy flux and the states of spatial coherence and polarization of random electromagnetic fields. So, the fundamental quantities of both the scalar generalised radiometry and the classical radiometry or photometry are deduced as particular cases of the tensor theory. A new procedure of analysis of (second-order) correlations, subject to the accomplishment of conservation laws, is also introduced. It reveals that (1) the primary sources of the measurable radiometric quantities associated to the random electromagnetic fields in any states of spatial coherence and polarization are the individual radiators of the radiant source (the correlations of the electric and magnetic field vectors only modulate the contributions given by those radiators) and (2) there are two physical mechanisms for the transport of measurable radiometric quantities by the electromagnetic field, i.e. the propagation of the contributions from individual radiators and their redistribution over each wavefront on propagation. The term redistribution refers to the transfer of portions of the measurable quantity over the wavefronts on propagation, without change its total value over each wavefront. In this context, a physical meaning is given to the negative values of the generalised radiance, which gives new insight about the Poynting’s theory of energy transport.     

Propagation of random electromagnetic beams through axially nonsymmetrical optical systems

When random electromagnetic beams passing through axially nonsymmetrical ABCD optical systems, the analytical formula for the transformation of the elements of 2 × 2 cross-spectral density matrix is obtained with the help of vector integration. We derive analytical expressions of the spectral degree of polarization, the spectral degree of coherence, and the spectral density in any output plane z > 0. Some numerical calculations are illustrated relating to the electromagnetic Gaussian Schell-model beams propagating through such optical systems.     

The symmetry properties of stable polynomial Gaussian beam profiles

We consider the symmetry properties of polynomial Gaussian beam profiles (intensity distributions) that remain stable during propagation, apart from being scaled and possibly rotated. These beams are expressed as special linear combinations of the Laguerre-Gaussian modes. Two kinds of symmetries are present: discreet rotational symmetries and mirror symmetries. The symmetry properties are shown to depend on the particular subset of Laguerre-Gaussian modes that is used to construct the stable beam. We demonstrate the symmetry properties of a few examples of stable beams through numerical simulations.     

Stokes parameters and degree of polarization of nonparaxial stochastic electromagnetic beams

On the basis of angular spectrum representation and the stationary-phase method, far-field expressions for generalized Stokes parameters of nonparaxial stochastic electromagnetic beams are derived, which permits us to study the changes in the ordinary Stokes parameters upon propagation, and the changes in the spectral degree of polarization of partially polarized nonparaxial stochastic electromagnetic beams. It is shown that the spectral degree of polarization changes across the section on beam propagation.     

Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere

With the help of the generalized Huygens–Fresnel integral and the ABCD matrix approach a bistatic LIDAR system involving a rough target at a distant location in a turbulent atmosphere is modeled. The system operates by means of an optical beam which has arbitrary spectral composition, and states of coherence and polarization. The rough target is modeled as a combination of a Gaussian mirror and a thin phase screen which induces phase perturbations of the components of the electric field. The analytical form of the cross-spectral density matrix of the returned beam is determined, from which the effect of the rough target on the spectral density (intensity) and polarization of the returned wave is analyzed. E. Watson’s work performed while assigned to the Ladar and Optical Communications Institute, University of Dayton, Dayton, OH 45469, USA.