Optical trapping Rayleigh dielectric particles with focused partially coherent dark hollow beams

The focusing properties of coherent and partially coherent dark hollow beams (DHBs) through a paraxial ABCD optical system are theoretically investigated. It is found that the evolution behavior of the intensity distribution of focused partially coherent DHBs is closely related to their spatial coherence. The radiation forces (RFs) of focused coherent and partially coherent DHBs acting on a Rayleigh dielectric particle are also theoretically investigated. Numerical results show that the coherent and partially coherent DHBs can be focused into a tight focal spot, which can be used to stably trap a Rayleigh dielectric particle with high refractive index at the focus point. The influences of different beam parameters, including the spatial coherence, beam waist width, beam order, and hollow parameter of partially coherent DHBs, on the RFs and the trap stiffness are analyzed in detail. Finally, the stability conditions for effective trapping particles are also discussed.     

Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere

Spectral changes of a partially coherent standard or elegant Laguerre–Gaussian (LG) beam propagating in turbulent atmosphere were studied numerically. Our results show that the spectral changes of a partially coherent standard or elegant LG beam in turbulent atmosphere are determined by both the structure constant of the turbulent atmosphere and the initial beam parameters. Furthermore, it is found that a partially coherent elegant LG beam is less affected by the turbulent atmosphere than a partially coherent standard LG beam from the aspect of the on-axis spectral shift, and this advantage is enhanced for small structure constant, small beam waist size, large mode orders, and large transverse coherence length. Our results will be useful in long-distance free-space optical communications.     

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.     

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.     

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.     

Propagation of a partially coherent higher-order cosh-Gaussian beam through a paraxial ABCD optical system

Based on the generalized Huygens–Fresnel integral, analytical expressions for the mutual coherence function, the spatial complex degree of coherence, and the effective size of a partially coherent higher-order cosh-Gaussian beam through a paraxial ABCD optical system have been derived. As a numerical example, the propagation of a partially coherent higher-order cosh-Gaussian beam through an optical Fourier-transforming system with a limiting aperture is illustrated. The normalized intensity distribution, the spatial complex degree of coherence, and the effective beam size for the partially coherent higher-order cosh-Gaussian beam are numerically demonstrated in the observation plane. The influences of the spatial coherence length and the limiting aperture on the normalized intensity distribution, the spatial complex degree of coherence, and the effective beam size are also examined in detail.     

Focal shift in focused truncated pulsed-laser beam

The focal shift of a focused truncated pulsed-laser beam is investigated. In the case of the Fresnel approximation, the analytic expression of the time-averaged intensity distribution along the axis is derived based on the series expansion. It shows that the focal shift of the pulsed beam can be completely determined by a series of normalized spectrum moments and the central Fresnel number defined according to the central frequency of the pulse. The absolute value of the focal shift of the pulsed beam decreases monotonously and slowly with the normalized spectrum width increasing and the central Fresnel number fixed, and it increases monotonously with the central Fresnel number decreasing and the normalized spectrum width fixed. Besides the central Fresnel number and the normalized spectrum width, the shape of spectral intensity of the pulse affects the focal shift too.     

Evolution properties of a partially coherent Lorentz-Gauss vortex beam in a uniaxial crystal

The propagation properties of a partially coherent Lorentz-Gauss vortex beam propagating in a uniaxial crystal orthogonal to the optical axis were studied. The analytical expressions of the partially coherent Lorentz-Gauss vortex beam propagating in a uniaxial crystal were derived. Based on the derived formulae, the average intensity properties and evolution properties of the coherent vortex of the partially coherent Lorentz-Gauss vortex beam propagating in a uniaxial crystal were illustrated using numerical examples. The influences of the uniaxial crystal and the parameters of the beam at the source plane of z?=?0 on the average intensity and evolution properties of the coherent vortex for the partially coherent Lorentz-Gauss vortex beam are analyzed in detail.     

Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere

Based on the generalized Huygens–Fresnel integral, propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere was investigated. Analytical propagation formulae were derived for the cross-spectral densities of partially coherent Lorentz and Lorentz–Gauss beams. As an application example, the focusing properties of partially coherent Gaussian, Lorentz and Lorentz–Gauss beams in a turbulent atmosphere and in free space were studied numerically and comparatively. It is found that the focusing properties of such beams are closely related to the initial coherence length and the structure constant of turbulence. By choosing a suitable initial coherence length, a partially coherent Lorentz beam can be focused more tightly than a Gaussian or Lorentz–Gauss beam in free space or in a turbulent atmosphere with small structure constant at the geometrical focal plane.     

Propagation properties of cylindrical sinc Gaussian beam

Abstract

We investigate the propagation properties of cylindrical sinc Gaussian beam in turbulent atmosphere. Since an analytic solution is hardly derivable, the study is carried out with the aid of random phase screens. Evolutions of the beam intensity profile, beam size and kurtosis parameter are analysed. It is found that on the source plane, cylindrical sinc Gaussian beam has a dark hollow appearance, where the side lobes also start to emerge with increase in width parameter and Gaussian source size. During propagation, beams with small width and Gaussian source size exhibit off-axis behaviour, losing the dark hollow shape, accumulating the intensity asymmetrically on one side, whereas those with large width and Gaussian source size retain dark hollow appearance even at long propagation distances. It is seen that the beams with large widths expand more in beam size than the ones with small widths. The structure constant values chosen do not seem to alter this situation. The kurtosis parameters of the beams having small widths are seen to be larger than the ones with the small widths. Again the choice of the structure constant does not change this trend.     

Propagation of a partially coherent beam from an unstable resonator through turbulent atmosphere

The influence of atmospheric turbulence on the propagation of a partially coherent beam from an unstable resonator was studied numerically. The resonant mode of the unstable resonator is obtained by iterative calculation using the Huygens–Fresnel formula. Also, using the extended Huygens–Fresnel integral, the intensity distribution of a propagating laser beam is calculated for different conditions. The influence of turbulence on the profile of partially coherent beams of an unstable resonator is studied. The effects of geometrical parameters of the resonator on the far-field beam profile are investigated. The results show that an unstable resonator with higher magnification has a superior far-field beam profile under partial coherency and turbulence conditions.     

Acousto-optic Conversion of Laser Beams Into Flat-top Beams

Abstract

We employ computer-optimized synthetic acousto-optic holograms to convert a laser beam (not necessarily Gaussian) into a nearly uniform-intensity partially coherent field in the Fourier plane. The width of the flat-top region can be controlled electronically in real time.     

Focal swift for a Gaussian beam: an experimental study

The beam radius along a focused, unapertured Gaussian beam was measured and used to calculate the dependence of the geometrical Fresnel number on the effective Fresnel number of the beam as it emerged from the focusing lens. The resulting data clearly demonstrate a focal shift away from the focal plane given by geometrical optics. The data agree very well with a theory due to Carter. This theory indicates that a significant focal shift occurs in these beams if the focusing lens is placed within the near field of the focal plane.     

Shaping the focal intensity distribution using spatial coherence

The intensity and the state of coherence are examined in the focal region of a converging, partially coherent wave field. In particular, Bessel-correlated fields are studied in detail. It is found that it is possible to change the intensity distribution and even to produce a local minimum of intensity at the geometrical focus by altering the coherence length. It is also shown that, even though the original field is partially coherent, in the focal region there are pairs of points at which the field is fully correlated and pairs of points at which the field is completely incoherent. The relevance of this work to applications such as optical trapping and beam shaping is discussed.     

Phase-only diffractive optical elements with subdiffraction-limited depth of focus

Abstract

Phase-only subdiffraction-limited depth of focus (DOF) elements are proposed and a relatively simple method is given which reduces the number of design parameters to four. A diffractive optical element with subdiffractive-limited DOF was experimentally demonstrated for the first time. Experimental results are in close agreement with numerical prediction. Several calculated examples show that the DOF for subdiffraction-limited elements can decrease by an amount, ranging from 37% to 71% compared to the DOF for a diffraction-limited element when ρ is chosen close to zero. The reduction is determined by the focal length of the element and the required distance between the primary peak (focal plane) and the secondary peak, both in units of wavelength. The latter, i.e. the peak-to-peak distance, determines the maximum applicable DOF. The influence of input laser beam parameters, such as phase distribution and beam size, on the focusing performance have also been investigated.     

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.     

Transverse superresolution technique involving rectified Laguerre-Gaussian LG(p)⁰ beams

A promising technique has been proposed recently [Opt. Commun. 284, 1331 (2011), Opt. Commun. 284, 4107 (2011)] for breaking the diffraction limit of light. This technique consists of transforming a symmetrical Laguerre-Gaussian LG(p)? beam into a near-Gaussian beam at the focal plane of a thin converging lens thanks to a binary diffractive optical element (DOE) having a transmittance alternatively equal to -1 or +1, transversely. The effect of the DOE is to convert the alternately out-of-phase rings of the LG(p)? beam into a unified phase front. The benefits of the rectified beam at the lens focal plane are a short Rayleigh range, which is very useful for many laser applications, and a focal volume much smaller than that obtained with a Gaussian beam. In this paper, we demonstrate numerically that the central lobe's radius of the rectified beam at the lens focal plane depends exclusively on the dimensionless radial intensity vanishing factor of the incident beam. Consequently, this value can be easily predicted.     

High-aperture diffraction of a scalar, off-axis Gaussian beam

A scalar treatment for Gaussian beams offset from the optic axis and then focused by a high-numerical-aperture lens is presented. Such a theory is required for describing certain types of Doppler microscopes, i.e., when the measurement is simultaneously performed by more than a single beam axially offset and then focused by a lens. Analytic expressions for the intensity in the focal region of the high-aperture lens are derived. From these expressions we calculate the intensity in the focal region with parameters of beam size, beam offset, and the numerical aperture of the lens. The relative location and variation of the intensity around the focal region are discussed in detail. We show that for small-diameter Gaussian beams the Strehl ratio increases above unity as the beam is offset from the optic axis. This is explained by the increase in the effective numerical aperture of the offset beam compared with the one collinear with the optic axis. From examining the focal distribution, we conclude that it rotates for small beam size and that increasing beam diameter causes the focused distribution to rotate and shear, i.e., to distort. We also show that the distortion of the distribution increases with increasing numerical aperture.     

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.