Beam shaping of focused partially coherent beams by use of the spatial coherence effect

We demonstrate that when a partially coherent beam with a Gaussian intensity distribution is focused by a lens, the desired partially coherent flat-topped intensity distribution or doughnut-shaped intensity distribution at the geometrical focus can be generated by choice of appropriate form of spectral degree of coherence. We provide a novel approach to beam shaping of a partially coherent beam and offer new schemes for their potential applications such as material processing, optical therapy, and optical tweezers.     

Beam wander of partially coherent Airy beams

The beam wander of a partially coherent Airy beam in a turbulent atmosphere was investigated. By using the extended Huygens–Fresnel integral, as analytical expression is derived for the second-order moment of a partially coherent Airy beam. Based on the theory proposed by Andrews, a general expression is obtained for the beam wander of a partially coherent Airy beam. With the help of the expression, various factors which impact on the beam wander are illustrated numerically. The results show that the beam wander of a partially coherent Airy beam decreases with the increase of the characteristic scale and the decrease of the coherent length or the exponent truncation factor. The value of the beam wander is a maximum when the exponent truncation factor is 0.63, no matter what the coherent lengths are. Our results provide an effective way to control the beam wander of a partially coherent Airy beam in practice.     

Average characteristics of partially coherent electromagnetic beams

Average characteristics of partially coherent electromagnetic beams are treated with the paraxial approximation. Azimuthally or radially polarized, azimuthally symmetric beams and linearly polarized dipolar beams are used as examples. The change in the mean squared width of the beam from its value at the location of the beam waist is found to be proportional to the square of the distance in the propagation direction. The proportionality constant is obtained in terms of the cross-spectral density as well as its spatial spectrum. The use of the cross-spectral density has advantages over the use of its spatial spectrum.     

Spectral changes and spectral switches of partially coherent beams focused by an aperture lens

Starting from the propagation law of partially coherent polychromatic light in the space-frequency domain, detailed numerical results and physical analysis are given to elucidate spectral changes and spectral switches at the geometrical focal plane of Gaussian Schell-model beams focused by an aperture lens. It is found that, in contrast to the aperture-induced spectral anomalies of spatially fully coherent polychromatic light, for partially coherent polychromatic light aperture diffraction plays an important role in spectral switching, but the truncation parameter, spectral correlation, and bandwidth all affect its spectral behavior.     

Radiation efficiency of partially coherent electromagnetic beams.

We present a general definition of the radiation efficiency of stationary electromagnetic fields and prove that it is bounded between zero and unity for beams of any state of coherence and polarization. The radiation efficiency may be interpreted as a measure of how directed the radiated fields are, and therefore it can be used to assess the allowed spatial coherence and intensity variations across a beam. We consider a class of partially coherent electromagnetic fields that were recently introduced in the literature and evaluate the radiation efficiencies for two particular examples, namely, the azimuthally polarized symmetric beams and the dipolar beams that are nearly linearly polarized in the central region. The results show that the radiation efficiency is fairly insensitive to the state of polarization and that it differs appreciably from unity for only small values of source and correlation widths.     

Spreading of partially coherent beams in random media

Some published computational work has suggested that partially coherent beams may be less susceptible to distortions caused by propagation through random media than fully coherent beams. In this paper this suggestion is studied quantitatively by examining the mean squared width of partially coherent beams in such media as a function of the propagation distance. The analysis indicates under what conditions, and to what extent, partially coherent beams are less affected by the medium.     

Propagation of partially coherent beams through particulate media

When propagating through particulate media, optical beams are degraded owing to scattering. We found that the ratio between the width of the distorted beam and the width of the initial beam decreases when the spatial coherence of the incident beam is reduced. These experimental observations are well described within the paraxial approximation of the transport theory.     

Spatial correlation properties and correlation vortices of partially coherent vortex beams diffracted by an aperture

Taking the Gaussian–Schell model vortex beam as a typical example of partially coherent vortex beams, the spatial correlation properties and correlation vortices of partially coherent vortex beams diffracted by an aperture are studied. It is shown that the off-axis displacement and spatial coherence affects the spectral degree of coherence. The number and position of correlation vortices depend on the off-axis displacement, spatial coherence, aperture truncation and propagation distance, where the effect of aperture diffraction on the correlation vortices is stressed. The number of correlation vortices decrease as the truncation parameter increases. The correlation vortices in the diffracted field result from the vortex embedded in partially coherent beams at the source plane rather than from the aperture diffraction. The correlation vortices in the diffracted field appear even when the vortex core is stopped by the aperture.     

Propagation-induced polarization changes in partially coherent optical beams

Propagation of a partially coherent optical beam inside a linear, nondispersive, dielectric medium is studied, taking into account the vector nature of the electromagnetic field. Propagation-induced polarization changes are studied by using the Gaussian-Schell model for the cross-spectral-density tensor. The degree of polarization changes with propagation and also becomes nonuniform across the beam cross section. The extent of these changes depends on the coherence radius associated with the cross-correlation function. For optical beams with symmetric spectra, the bandwidth of the source spectra is found to play a relatively minor role.     

Optical superresolution of focused partially spatially coherent laser beams

We report design theories of a diffractive superresolution element (DSE) to implement optical superresolution of focused partially spatially coherent laser beams. The design problem of the DSE can be transformed into a problem of linear programming to obtain a globally optimal solution. By using the design theories, some fundamental limits of optical superresolution of focused partially spatially coherent laser beams are proposed, and several design examples are provided. As expected, both the fundamental limits and the design examples show that worse spatial coherence will cause worse superresolution performance. The design theories provide a design approach with partially coherent beams and may be useful for other design problems under partially coherent illumination.     

Statistics of partially coherent beams: a numerical analysis

We show that by using Monte Carlo simulations, one can study the characteristics of beams with adjustable spatial coherence properties that propagate through highly scattering media. Moreover, we show that a single simulation is sufficient to obtain the intensity distribution at the exit surface of the scattering medium for any degree of global coherence of the input beam. The efficient numerical procedure correctly reproduces the first- and second-order statistics of the intensity distribution obtained after propagation through diffusive media.     

A class of partially coherent beams carrying optical vortices

A new class of partially coherent beams with a separable phase, which carry optical vortices, is introduced. It is shown that any member of the class can be represented as an incoherent superposition of fully coherent Laguerre-Gauss modes of arbitrary order, with the same azimuthal mode index. The free-space propagation properties of such partially coherent beams are studied analytically, and their M2 quality factor is investigated numerically.     

Spatial correlation properties of focused partially coherent vortex beams

The spatial correlation properties in the geometrical focal region of a converging, partially coherent vortex wave field are analyzed. Expressions are derived for a pair of points on the axis of symmetry and for a pair of points in the focal plane. It is found that the longitudinal and transverse coherence lengths in the focal region change with the variation of the topological charge and the normalized coherence length of the vortex field. In addition, the degree of coherence is shown to possess phase singularities.     

Anisotropic pure-phase plates for quality improvement of partially coherent,partially polarized beams

From a theoretical point of view, the use of anisotropic pure-phase plates (APP) is considered in order to improve the quality parameter of certain partially coherent, partially polarized beams. It is shown that, to optimize the beam-quality parameter, the phases of the two Cartesian components of the field at the output of the APP plate should be identical and should exhibit a quadratic dependence on the radial polar coordinate.     

Effect of dispersion on the spectrum of partially coherent beams

Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, the analytical expressions of the spectrum of GSM beams propagating in dispersive media are derived, and the spectral properties are studied in detail. It is shown that, in comparison with propagation in free space and in turbulence, whether or not GSM beams satisfy the scaling law, the normalized spectrum of GSM beams in dispersive media changes on propagation in general, because the dispersive medium affects different spectral components differently. As compared with the free-space propagation, for the scaling-law GSM beams the dispersion results in spectrum change, and for the nonscaling-law GSM beams the dispersion gives rise to a further increase in spectral changes. The structure constant of the dispersive property of the media, the transverse coordinate of the observation point, the spatial correlation length of the source, and the propagation distance affect the spectral behavior of GSM beams; this effect is illustrated numerically.     

Propagation of partially coherent flat-topped beams through a turbulent atmosphere

Based on the modified beam model for flat-topped beams and the Schell model for partially coherent light, an expression for partially coherent flat-topped (PCFT) beams has been proposed. The propagation characteristics of PCFT beams with circular symmetry through a turbulent atmosphere have been studied. By using the generalized Huygens-Fresnel integral and Fourier transform method, the expressions for the cross-spectral density function and the average intensity have been given and the analytical expression for the root-mean-square width has been derived. The effects of the beam order, the spatial coherence, and the turbulent parameter on the intensity distributions and beam spreading have been discussed in detail. Our results show that the on-axis intensity of the beams decreases with increasing turbulence and decreasing coherence of the source, whereas the on-axis intensity of the beams in the far field decreases slightly with increasing beam order. The relative spreading of PCFT beams is smaller for beams with a higher order, a lower degree of global coherence of the source, a larger inner scale, and a smaller outer scale of the turbulence.     

Modal decomposition of partially coherent beams using the ambiguity function

Phase-space representations of optical beams such as the ambiguity function or the Wigner distribution function have recently gained considerable importance for the characterization of coherent and partially coherent beams. There is growing interest in beam properties such as the beam propagation factor and the coherence and phase information that can be extracted from these phase-space representations. A method is proposed to decompose a partially coherent beam into Hermite-Gaussian modes by using the ambiguity function. The modal weights and the possible phase relations of the Hermite-Gaussian modes are retrieved. The method can also be applied for the decomposition of the Wigner distribution function. Some examples are discussed in the scope of beam characterization.     

Generation of partially coherent diffraction-free fields with tunable geometry

The synthesis of diffraction-free fields with different profiles is described. The synthesis is done by illuminating a screen containing a circular slit with a cosine beam. The treatment is equivalent to the modulation of the slit transfer characteristics and makes possible a tunable interference interaction of Bessel beams with noncommon axes. These results are generalized, and it is shown that coherent diffraction-free fields with arbitrary profiles can be expressed as the superposition of shifted zero-order Bessel beams and temporary parametric representations of the shifted functions. Diffraction-free fields with partially coherent features can be obtained. Experimental results are shown for each case.     

Focal shift and focal switch of partially coherent flat-topped beams passing through an alignment and misalignment lens system with aperture

The focal shift and focal switch of partially coherent flat-topped (PCFT) beams passing through a lens system with an aperture are studied in detail. We have shown that a focal shift is also present for beams propagating through an aligned optical system and the amount of the focal shift depends not only on the radius aperture of the focusing system, but also on the spatial coherence and order of the flat-topped beam of the incident partially coherent light. A new phenomenon called ‘focal switch’ occurs for misaligned optical systems, i.e. the focal shift experiences a sudden transition as the aligned optical system becomes misaligned, and the influence of the spatial coherence, order of flat-topped beam and aperture size reduction, on the focal switch is investigated in detail. Finally, the necessary conditions for the focal switch to take place are investigated.     

Effects of atmosphere on spectrum of modified partially coherent flat-topped beams

The analytical formulas for the spectrum of modified partially coherent flat-topped (MPCFT) beams propagating in a turbulent atmosphere have been derived under strong fluctuation condition of turbulence. The spectral properties of MPCFT beams propagating in a strong turbulence have been investigated, and the changes of on-axis and off-axis spectral distributions have been analyzed quantitatively. The results show that the on-axis relative spectral shift of the modified partially coherent flat-topped beam exhibits spectral switch for the beam order M > 1, and with the increasing propagation distance the blue-shift gradually decreases in atmospheric turbulence. A rapid transition for the off-axis spectrum appears at one critical position in turbulence, and the position decreases with the increasing beam order for the lower beam order and the nearer propagation distance. The spectral properties and the spectral shifts of MPCFT beams also depend on the inner scale, the spatial coherence parameter.