Study Of Some Effects In Coherent And Correlation Singular Optics

Singular optics recognized as an important branch of modern optics has been recently developed rapidly, which has been extended from primarily dealing with fully coherent, polychromatic wavefields (called coherent singular optics) to treating partially coherent, quasi-monochromatic wavefields (called correlation singular optics), and fully coherent, polychromatic wavefields. It is well known that phase singularities have unique and fascinating features which are ubiquitous throughout the wavefields. Based on the comparison of the application of the diffraction integral formulae in this dissertation, some effects in coherent and correlation singular optics are studied. The major contents and results are as follows:Starting from the vectorial Rayleigh-Sommerfeld (RS) diffraction integrals, the analytical expressions of vectorial Gaussian beams through an annular aperture are derived. The on-axis field and far field expressions of vectorial nonparaxial Gaussian beams diffracted at the annular aperture, the nonparaxial vector Gaussian beams diffraction at a circular aperture and a disk, as well as the free-space propagation equation of nonparaxial vector Gaussian beams are treated in a unified way as special cases of our general result. Numerical comparative examples are given to illustrate the intensity distributions and far-field behavior of nonparaxial vector Gaussian beams through an annular and a circular disk aperture. The validity of our treatment is confirmed by direct numerical integration of the RS formulae. It is shown that both the f-parameter and annular obscuration (or truncation parameter) affect the beam nonparaxiality in the case of diffraction at the annular (or disk) aperture. Based on the vectorial RS diffraction integral formula and using a simple expansion of the nucleus in the RS formula, an analytical propagation equation of vectorial nonparaxial Gaussian beams in free space is given, which permits us to perform numerical calculations in comparison with the expression derived by Ciattoni et al. and with the direct numerical integration of the RS formula. It is found that as usual the use of expansion of vectorial RS diffraction integral is sufficient to provide satisfactory numerical results as compared with the direct integration of the RS formula. The above two analytical expressions are valid under certain conditions, however, both are applicable in the far field.The equivalence of the vectorial angular-spectrum representation and RS diffraction integral formulae is studied. Based on the angular-spectrum representation and the Weyl representation of a spherical wave, the vectorial RS diffraction formulae of the first and second kinds are derived in a simple way. Numercial results of diffracted divergent spherical waves are given to illustrate the application of the two vectorial RS diffraction formulae.The integral expression for divergent spherical waves diffracted at an annular aperture is derived based on the Maggi-Rubinowic theory of the boundary diffraction wave and Helmhotz-Kirchhoff integral theorem. The expressions for divergent spherical waves diffracted at a circular aperture and a disk, and the axial field are treated as the special cases of our general one. Numerical calculation results for axial and transversal intensity distributions are given to compare our results with the Kirchhoff diffraction integral, first and second RS diffraction integrals. It is shown that the results with the boundary diffraction wave are in agreement with those in the use of the Kirchhoff diffraction integral, but the computer time is reduced greatly by using the boundary diffraction wave theory. The four diffraction formulae are shown to be consistent for axial and transversal intensity distributions, if the source and observation points are positioned equally from the aperture, or the observation point is located enough far from the aperture. Otherwise, the mean value of the first and second RS diffraction integrals is equal to the result of the boundary diffraction wave theory. Taking the Gaussian background vortex beam with topological charge +2 as a typical example, closed-form expressions for vortex Gaussian beams in free space and passing through a half-plane screen are derived for the first time. Numerical examples are given to study the propagation dynamics of on-axis and off-axis vortex beams in detail. It is shown that in the free-space propagation of off-axis vortex Gaussian beams there is and only is one phase singularity which rotates about the z axis, and the orientation of rotation of the vortex depends on the sign of topological charge. The rotating angleφincreases with increasing propagation distance z, andφ→90°as z→+∞. The topological charge is conserved during the free-space propagation. For the case of the half-plane diffraction of vortex Gaussian beams there may exist more than one phase singularities or no phase singularity in the diffraction field. The number and position of phase singularities are dependent on the vortex position at the source plane and propagation distance. The creation, motion and annihilation of phase singularities in the diffraction field may appear by varying the vortex position and propagation distance. Additionally, the total topological charge of the vortex diffracted Gaussian beam is zero in the diffraction field which is not equal to the topological charge of the phase singularity at the source plane. It means that the change of topological charge may result from the half-plane diffraction.The creation-annihilation process of phase singularities of flat-topped beams in the focal plane is studied within the framework of the scalar paraxial approximation, where the new expression for flat-topped beams recently introduced by Li is taken as the beam model. It is shown that by suitably varying the truncation parameter, the phase singularities reorganize themselves. The results are similar to those of scalar paraxial Gaussian beams where the annihilation of singularities in the focal plane, the creation of singularities outside the focal plane and subwavelength structures appear. During the reorganization process the total topological charge is conserved. However, apart from the truncation parameter, the beam order of flat-topped beams additionally affects the spatial distribution of phase singularities.By using the vectorial Debye diffraction theory, phase singularities of high numerical aperture (NA) dark-hollow Gaussian beams in the focal region are studied for the first time. The dependence of phase singularities on the truncation parameter and semi-aperture angle a (or equally, NA) is illustrated numerically. A comparison of phase singularities of high NA dark-hollow Gaussian beams with those of scalar paraxial Gaussian beams and high NA Gaussian beams is made. For high NA dark-hollow Gaussian beams the beam order n additionally affects the spatial distribution of phase singularities, for example, the innermost n singularities in focal plane do not disappear in the creation and annihilation process. In addition, by variation of the semi-aperture angle in a certain region, there exist phase singularities outside the focal plane, which may be created and annihilated.The coherence vortices in a new class of partially coherent beams with separable phase recently proposed by Bogatyryova et al. are studied in detail. The new beams are generated by taking an incoherent superposition of Laguerre-Gaussian (LG) modes with equal azimuthal index. It is shown that the mode indices and weight factors of superposed LG modes, as well as the reference choice affect the position where the circular edge dislocation takes place, and result in the coherence vortex disappearance or the appearance of more than one coherence vortices. Furthermore, the coherence vortices of partially coherent beams with non-separable phase consisting of an incoherent superposition of LG modes with unequal azimuthal index are studied for the first time. Detailed numerical examples are given to illustrate the coherence vortices which are no longer circular edge dislocations in the transverse plane, but the mode indices and weight factors of superposed LG modes, and the reference choice also affect the number and positions of the coherence vortices.By using the generalized Debye diffraction integral, the spatial correlation properties and phase singularity annihilation of apertured Gaussian Schell-model (GSM) beams in the focal region are studied for the first time. It is shown that the width of the spectral degree of coherence can be larger, less than or equal to the corresponding width of spectral density, which depends not only on the scalar coherence length of the beams, but also on the truncation parameter. With a gradual increase of the truncation parameter, a pair of phase singularities of the spectral degree of coherence in the focal plane approaches each other, resulting in subwavelength structures. Finally, the annihilation of pairs of phase singularities takes place at a certain value of the truncation parameter. With increasing scalar coherence length, the annihilation occurs at the larger truncation parameter. However, unlike the case of fully coherent apertured Gaussian beams, during the annihilation process no creation of phase singularities outside the focal plane is found for GSM beams.Based on the propagation law of cross-spectral density function, the coherence vortices of partially coherent, quasi-monochromatic singular beams with Gaussian envelope and Schell-model correlator in the far field are studied, where our main attention is paid to the evolution of far-field coherence vortices into intensity vortices of fully coherent beams. It is shown that, although there are usually no zeros of intensity in partially coherent beams with Gaussian envelope and Schell-model correlator, zeros of spectral degree of coherence exist. The coherence vortices of spectral degree of coherence depend on the relative coherence length, modex index and positions of pairs of points. If a point and mode index are kept fixed, the position of coherence vortices changes with an increase of the relative coherence length. For the low coherent case there is a circular phase dislocation. In the coherent limit coherence vortices become intensity vortices of fully coherent Laguerre-Gaussian beams.In summary, the results presented in this dissertation studying on some effects in coherent and correlation singular optics are further enrichment and extension of the research range of singular optics, and can be useful for applications of optical vortices field.