Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane

An analytical and concise formula is derived for the fractional Fourier transform (FRT) of partially coherent beams that is based on the tensorial propagation formula of the cross-spectral density of partially coherent twisted anisotropic Gaussian-Schell-model (GSM) beams. The corresponding tensor ABCD law performing the FRT is obtained. The connections between the FRT formula and the generalized diffraction integral formulas for partially coherent beams passing through aligned optical systems and misaligned optical systems are discussed. With use of the derived formula, the transformation and spectrum properties of partially coherent GSM beams in the FRT plane are studied in detail. The results show that the fractional order of the FRT has strong effects on the transformation properties and the spectrum properties of partially coherent GSM beams. Our method provides a simple and convenient way to study the FRT of twisted anisotropic GSM beams.     

Experimental observation of truncated fractional Fourier transform for a partially coherent Gaussian Schell-model beam

The truncated fractional Fourier transform (FRT) is applied to a partially coherent Gaussian Schell-model (GSM) beam. The analytical propagation formula for a partially coherent GSM beam propagating through a truncated FRT optical system is derived by using a tensor method. Furthermore, we report the experimental observation of the truncated FRT for a partially coherent GSM beam. The experimental results are consistent with the theoretical results. Our results show that initial source coherence, fractional order, and aperture width (i.e., truncation parameter) have strong influences on the intensity and coherence properties of the partially coherent beam in the FRT plane. When the aperture width is large, both the intensity and the spectral degree of coherence in the FRT plane are of Gaussian distribution. As the aperture width decreases, the diffraction pattern gradually appears in the FRT plane, and the spectral degree of coherence becomes of non-Gaussian distribution. As the coherence of the initial GSM beam decreases, the diffraction pattern for the case of small aperture widths gradually disappears.     

Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics

We report the experimental observation of the fractional Fourier transform (FRT) for a partially coherent optical beam with Gaussian statistics [i.e., partially coherent Gaussian Schell-model (GSM) beam]. The intensity distribution (or beam width) and the modulus of the square of the spectral degree of coherence (or coherence width) of a partially coherent GSM beam in the FRT plane are measured, and the experimental results are analyzed and agree well with the theoretical results. The FRT optical system provides a convenient way to control the properties, e.g., the intensity distribution, beam width, spectral degree of coherence, and coherence width, of a partially coherent beam.     

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.     

Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems

Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned ABCD optical systems is investigated based on the generalized Collins formula for treating the propagation of a partially coherent beam through such optical systems. Analytical formulas for the cross-spectral density of a partially coherent Hermite-Gaussian beam propagating through such optical systems are derived. As an application example, we derive the propagation formulas for a partially coherent flattened Gaussian beam by expressing it as a superposition of a series of partially coherent Hermite-Gaussian beams by using polynomial expansion. The focusing properties of a partially coherent Hermite-Gaussian beam focused by a thin lens are studied as a numerical example.     

Effect of turbulence on the beam quality of apertured partially coherent beams

The effects of turbulence on the beam quality of apertured partially coherent beams have been studied both analytically and numerically. Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, closed-form expressions for the average intensity, mean-squared beam width, power in the bucket, beta parameter, and Strehl ratio of apertured partially coherent beams propagating through atmospheric turbulence are derived. It is shown that the smaller the beam truncation parameter is, the less affected by turbulence the apertured partially coherent beams are. Furthermore, the apertured partially coherent beams are less sensitive to the effects of turbulence than unapertured ones. The main results are interpreted physically.     

Coherent and incoherent off-axis Hermite-Gaussian beam combinations

A detailed study of the coherent and the incoherent combinations of two-dimensional off-axis Hermite-Gaussian beams with rectangular symmetry is made. The closed-form propagation formulas of the resulting beam are derived, and the resulting beam quality in terms of the M(2) factor and power in the bucket is discussed and compared for the coherent and the incoherent combinations. In addition, it is shown that the resulting astigmatic beam can be symmetrized in the sense of the second-moment definition of beam width. However, the symmetrizing transformation of the resulting astigmatic beams is incomplete, because there exist different irradiance profiles.     

M2-factor of truncated partially coherent beams propagating through atmospheric turbulence

A method of studying the M2-factor of truncated partially coherent beams both in free space and in turbulence is proposed, i.e., the method of the window function being expanded into a finite sum of complex-valued Gaussian functions. Taking the Gaussian Schell-model (GSM) beam as a typical example of partially coherent beams, the analytical formula of the M2-factor of truncated GSM beams propagating through atmospheric turbulence is derived. It is shown that the M2-factor decreases as the truncation parameter δ and the coherence parameter α increase. However, the M2-factor in turbulence is more sensitive to δ than that in free space. On the other hand, the M2-factor of truncated partially coherent beams with smaller δ is more affected by turbulence. In addition, the effect of turbulence on the M2-factor of truncated GSM beams is less sensitive to the coherence parameter α than that of nontruncated GSM beams.     

Partially coherent flat-topped beam and its propagation

A partially coherent beam with flat-topped profile is proposed. The cross-spectral density of this beam can be expressed as a finite series of the cross-spectral density of partially coherent Gaussian-Schell-model beams with different parameters. Analytical propagation formulas for partially coherent flat-topped beams are derived through aligned and misaligned optical systems. The propagation property of partially coherent flat-topped beams in free space is illustrated numerically. The fractional Fourier transform of partially coherent fiat-topped beams is also studied. Our method provides a convenient way to describe partially coherent flat-topped beams and treat their propagation and transformation.     

Scattering of a partially coherent Gaussian-Schell beam from a diffuse target in slant atmospheric turbulence

On the basis of the extended Huygens-Fresnel principle, the scattering of partially coherent Gaussian-Schell-model (GSM) beams from a diffuse target in slant double-passage atmospheric turbulence is studied and compared with that of fully coherent Gaussian beams. Using the cross-spectral density function of the GSM beams, we derive the expressions of the mutual coherence function, angle-of-arrival fluctuation, and covariance and variance of the intensity of the scattered field, taking into account the fluctuations of both the log-amplitude and phase. The numerical results are presented, and the influences of the wavelength, propagation distance, and waist radius on scattering properties are discussed. The perturbation region of the normalized intensity variance of the partially coherent GSM beam is smaller than that of the fully coherent Gaussian beam at the middle turbulence level. The normalized intensity variance of long-distance beam propagation is smaller than that of beam propagation along a short distance.     

Generation of partially coherent laser beam directly from spatial-temporal phase modulated optical resonators

Abstract

The generation of a partially coherent laser beam directly from a spatial-temporal phase modulated optical resonator is investigated both experimentally and theoretically. The laser material used in the experiment is Nd:YAG rod pumped by Krypton lamps working in continuous wave mode. The phase modulation is fulfilled by an intra-cavity LiNbO₃ electro-optic crystal driven by high voltage. The experimental results show that intracavity phase modulation is an effective way to generate partially coherent laser beams. The theoretical analysis and numerical simulation shows that the output beam can be characterized by Gaussian Schell-model (GSM) beams. The two-slit interference experiment confirms that the output beam is partially coherent.     

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.     

Theoretical study on the spectral shifts of partially coherent array beams passing through ABCD optical systems

According to the extended Huygens–Fresnel integral, the expressions for the on-axis spectrum of partially coherent Gaussian Schell-model (GSM) rectangular array beams passing through ABCD optical systems have been derived. The generalized Fresnel number of the system, the spatial coherent parameter of array beamlets and the array beam parameters including the number of beamlets and the separation distance between beamlets have been taken as the characteristic parameters to compare the spectral shifts of GSM array beams for the two types of the superposition, i.e. the correlated superposition and the uncorrelated superposition. In particular, the effect of characteristic parameters on the on-axis relative spectral shifts has been discussed in detail. The results show that the spectral intensity of GSM array beams for the two types of the superposition passing through ABCD systems depends on the source spectral density S ⁰(ω), the spatial coherent parameter of array beamlets β, the generalized Fresnel number of the system F and the array beam parameters. Furthermore, for the uncorrelated superposition, the spectrum of GSM array beams only exhibit the blue-shift, whereas for the correlated superposition, GSM array beams exhibit the spectral switch and the number of spectral switches increases with the increase of array beam parameters in the near field due to the interference between beamlets. In particular, the effect of the array beam parameters on the on-axis relative spectral shift is more obvious.     

Analysing the behaviour of partially coherent divergent Gaussian beams propagating through oceanic turbulence

Theoretical study of propagation behaviour of partially coherent divergent Gaussian beams through oceanic turbulence has been performed. Based on the previously developed knowledge of propagation of a partially coherent beam in atmosphere, the spatial power spectrum of the refractive index of ocean water, extended Huygens–Fresnel principle and the unified theory of coherence and polarization, analytical formulas for cross-spectral density matrix elements are derived. The analytical formulas for intensity distribution, beam width and spectral degree of coherence are determined by using cross-spectral density matrix elements. Then, the effects of some source factors and turbulent ocean parameters on statistical properties of divergent Gaussian beam propagating through turbulent water are analysed. It is found that beam’s statistical propagation behaviour is affected by both environmental and source parameters variations.     

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.     

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.     

Coincidence fractional Fourier transform implemented with partially coherent light radiation

We introduce the coincidence fractional Fourier transform (FRT) implemented with incoherent and partially coherent light radiation. Optical systems for implementing the coincidence FRT are designed. The results show that the visibility and quality of the coincidence FRT of an object are closely related to the light source's transverse size, coherence, and spectral width. As an example, we numerically study the coincidence FRT of a single slit.     

Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media

By adopting a new tensor method, we derived an analytical propagation formula for the cross-spectral density of partially coherent twisted anisotropic Gaussian Schell-model (GSM) beams through dispersive and absorbing media. Using the derived formula, we studied the evolution properties and spectrum properties of twisted anisotropic GSM beams in dispersive and absorbing media. The results show that the dispersive and absorbing media have strong influences on the propagation properties of twisted anisotropic GSM beams and their spectrum evolution. Our method provides a simple and convenient way to study the propagation of twisted anisotropic GSM beams in media with complex refractive index.     

Partially coherent flattened Gaussian beam and its paraxial propagation properties

A simple model called partially coherent flattened Gaussian beam (FGB) is proposed to describe a partially coherent beam with a flat-topped spatial profile. An explicit and analytical formula is derived for the cross-spectral density of a partially coherent FGB propagating through a paraxial ABCD optical system. The propagation factor and propagation properties of a partially coherent FGB in free space are studied in detail and found to be closely related to its coherence and beam order.