Diffraction Integral Theory Research Articles

Tightly focused optical skyrmions and merons formed by electric-field vectors with prescribed characteristics

Abstract Optical skyrmions, which are topological quasi-particles with nontrivial electromagnetic textures, have garnered escalating research interest recently for their potential in diverse applications. In this paper, we present a method for generating tightly focused optical skyrmion and meron topologies formed by electric-field vectors under 4π-focusing system, where both the topology types (including Néel-, Bloch-, intermediate- and anti-skyrmion/meron) and the normal direction of the two-dimensional topology projection plane can be tailored at will. By utilizing time-reversal techniques, we analytically derive the radiation pattern of a multiple concentric-ring array of dipoles (MCAD) to obtain the required illumination fields on the pupil planes of the two high numerical aperture lenses. The Deby vector diffraction integral theory is employed to calculate the corresponding tightly focused field, and their topology characteristics are quantitatively evaluated by the electric-field vector distribution. The results demonstrate that arbitrary electric-field based skyrmion and meron can be conveniently generated by adjusting the oscillation direction of each dipole in the MCAD and the normal direction of the dipole array. The generated optical topologies with fully controllable degrees of freedom provide potential applications in optical information processing, transmission, and storage.

Deep learning-enhanced fluorescence microscopy via confocal physical imaging model.

Confocal microscopy is one of the most widely used tools for high-resolution cellular, tissue imaging and industrial inspection. Micrograph reconstruction based on deep learning has become an effective tool for modern microscopy imaging techniques. While most deep learning methods neglect the imaging process mechanism, which requires a lot of work to solve the multi-scale image pairs aliasing problem. We show that these limitations can be mitigated via an image degradation model based on Richards-Wolf vectorial diffraction integral and confocal imaging theory. The low-resolution images required for network training are generated by model degradation from their high-resolution counterparts, thereby eliminating the need for accurate image alignment. The image degradation model ensures the generalization and fidelity of the confocal images. By combining the residual neural network with a lightweight feature attention module with degradation model of confocal microscopy ensures high fidelity and generalization. Experiments on different measured data report that compared with the two deconvolution algorithms, non-negative least squares algorithm and Richardson-Lucy algorithm, the structural similarity index between the network output image and the real image reaches a high level above 0.82, and the peak signal-to-noise ratio can be improved by more than 0.6 dB. It also shows good applicability in different deep learning networks.

Statistical Properties of Partially Coherent Higher-Order Laguerre-Gaussian Power-Exponent Phase Vortex Beams

In this paper, partially coherent radially polarized (RP) Laguerre-Gaussian (LG) rotationally symmetrical power-exponent phase vortex (RSPEPV) beams with the LG-correlated Schell-model (LGSM) were introduced. The statistical properties of the tightly focused beams, including intensity distribution, degrees of polarization and coherence, and Stokes vector, were studied based on vectorial Richards-Wolf diffraction integral theory. Moreover, when the distance between focal plane and the observation plane z = 0, the relationships between the tight-focusing properties of RP-LG-RSPEPV beams with LGSM and the order of LGSM p’, topological charges l, power exponent n, spatial correlation δ, and radial index p were investigated. The results show that by changing the order of LGSM, topological charge, power exponent, spatial correlation length, and radial index, the focal spot distribution of various shapes can be obtained. This work provides ideas for the application of partially coherent beams in particle capture and optical tweezers.

Experimental generation and orbital angular momentum properties of focused field for vector vortex beams on Hybrid-order Poincaré sphere

Due to the anisotropic spatial polarization state and spiral phase distribution, the orbital angular momentum (OAM) of vector vortex beams (VVBs) has attracted increasingly attention in recent years. In this paper, the Hybrid-order Poincaré sphere VVBs are constructed by orthogonal left-handed (L-H) and right-handed (R-H) circularly polarized vortexes. The vector vortex light fields of different orders at the focal plane are collected and analyzed experimentally. Based on the vector diffraction integral theory, the effects of the topological charge and polarization order for VVBs, as well as the initial phase difference for L-H and R-H components on the OAM properties of focused field for the longitudinal component are investigated numerically. It is found that the OAM density distribution of the longitudinal component shows inner and outer two layers on the whole, and the layers mainly concentrate the corresponding L-H and R-H components respectively. Meanwhile, the signs of the topological charges for L-H and R-H components determine the positive and negative distributions of the inner and outer OAM density respectively. When the initial phase difference of L-H and R-H components ϕ0 changes in the range of 0–2π, the OAM density rotates the angle ϕ0/2 along the clockwise. However, the OAM of the longitudinal component does not change with ϕ0, and increases as the polarization order m increases when the topological charge of VVBs l>2. The results could be helpful to further explore the applications of the OAM for VVBs in the fields of optical detection and communication.

Focusing characteristics of linearly polarized ultrashort pulses at the focal plane.

The dynamic focusing characteristics of linearly polarized ultrashort pulses are studied. Both the complex source-sink model (CSSM) and the Richards-Wolf diffraction integral theory (RWT) are used to study the focusing phenomena. For the central focus spot, the descriptions of both the CSSM and the RWT are well consistent. Also, the CSSM can describe the super-resolution focused spot very conveniently, and only the beam waist parameters need to be changed. The dynamic convergence and divergence focusing phenomena of linearly polarized ultrashort pulse are studied by both the CSSM and RWT. The numerical simulation results of both the CSSM and the RWT are not consistent. In the convergent focusing process, there are dynamic focusing phenomena transitions from the halo to two light lobes to the elliptical focus spot. In the divergent defocusing process, the phenomena are the inverse process of the phenomena in the focusing process. The peak power of halos versus the beam convergence angles are studied. The specific angles corresponding to the significantly reduced peak powers of halos are given. These studies may be applied in the field of particle manipulation and acceleration.

Convergence and divergence focusing phenomena at the focal plane of ultrashort pulses.

Using the Richards-Wolf diffraction integral theory and the tightly focused ultrashort pulse vector model, the focusing phenomena at the focal plane of subcycle and few-cycle radially polarized ultrashort pulses are studied. The dynamic focusing is revealed at the focal plane. First, the subcycle or few-cycle ultrashort pulses shrink towards the focus. Then the ultrashort pulses diverge from the focus. So, the convergence and divergence moving halo at the focal plane can be observed. When approaching the focus, the amplitude of the pulse becomes larger. The phenomena can be understood from the Huygens-Fresnel principle and are important for applications of the focused ultrashort pulses.

Simultaneous identification of the azimuthal and radial mode indices of Laguerre-Gaussian beams using a spiral phase grating

Transfer of information using optical vortex beams as signal carriers can extremely extend the capacity and spectral efficiency in optical communication systems. One of the key problems is to accurately measure the mode indices including radial mode p and topological charge l at the receiver side of an optical vortices transmission system. In this paper, we present a new scheme to determine the mode indices of optical vortices by using spiral phase gratings. Based on the Fresnel diffraction integral theory, the far-field diffraction intensity distributions of vortex beams through the gratings are numerically calculated. Simulation results show that the annular-shaped intensity profiles of incident vortex beams evolve to bright spot array diffraction patterns, which render the information of the mode indices. Specifically, the magnitude of topological charge l and radial index p can be easily determined by counting the number of the corresponding bright stripes. And the orientation of the diffraction optical fields reveals the sign of the topological charge. Experiments are implemented to demonstrate the feasibility and validity of the proposed detecting method by uploading such phase gratings on a reflective phase-only liquid crystal spatial light modulator. The experimental observations are in good agreement with the theoretical predictions.

Investigating the nonlinear optical properties of poly eosin-Y phthalate solution under irradiation with low power visible CW laser light

In this work, the study of the nonlinear optical properties of poly eosin-Y phthalate solution using the diffraction ring patterns and the Z-scan methods are presented. The diffraction ring patterns were observed when a low power laser beam of wavelength of 473 nm passed through the poly eosin-Y phthalate solution. It was found that the number of rings and diameter of the outer-most ring in each pattern depends on the incident intensity of the laser beam and type of the beam wave front. The Fresnel-Kirchhoff diffraction integral theory was used to simulate the diffraction ring patterns and found good agreement with experimental results. Both diffraction ring patterns and closed aperture Z-scan techniques were used to calculate the nonlinear refractive index, while open aperture Z-scan was used to calculate the nonlinear absorption coefficient of the poly eosin-Y phthalate solution. The possibility of using the sample as an optical limiter is tested.

Hollow-Bottle microstructure beam shape transformation based on the confocal optical system

In this study, a confocal optical system formed by a combination of two axicon lenses, a parabolic annular mirror and a concave mirror was used to obtain an adjustable microstructured beam. The two axicons were used to create a collimated hollow annular beam whose diameter varies with the distance between the two components, and the duty cycle ε of the hollow beam changes accordingly. At the same time, the annular factor δ also changes correspondingly after the hollow beam was concentrated. According to the scalar diffraction integral theory, the change of focusing characteristics of different annular factor δ after a parabolic annular mirror was studied. The concentrated beam was reflected by a concave mirror to form a confocal optical system. According to the numerical simulation, the size of the focused spot varies from 2.8 to 4.3μm in the range of the annular factor δ from 1.19 to 1.09. Detailed characteristics of the microstructure beam shape changes were analyzed. These results have important implications for particle manipulation and other applications.

Far-field diffraction patterns and optical limiting properties of bisdemethoxycurcumin solution under CW laser illumination

In this work, nonlinear optical properties of bisdemethoxycurcumin solution were studied using diffraction ring patterns method. Multiple diffraction rings observed as a continuous wave (CW) laser beam passes through a bisdemethoxycurcumin solution. The nonlinear refractive index, n2, and change in refractive index, Δn, for sample were determined from the number of observed rings, and are found to be 0.8327 × 10−6 cm2/W and 0.00946 respectively. The evolution of the diffraction ring patterns are numerically studied using the Fresnel–Kirchhoff diffraction integral theory. The experimental findings are in good agreement with theoretical analysis. Also the optical limiting properties of the sample were measured at 473 nm. The obtained results proved that the sample bear good optical limiting property.

Arbitrarily spin-orientated and super-resolved focal spot.

In this Letter, we propose a facile approach for achieving a robust focal spot bearing both super-resolution and arbitrary spin orientation. Toward this aim, we meticulously devise a structured incident light consisting of three sorts of beams, which can be produced definitely by the superposition of a radially polarized beam and an azimuthally polarized beam. Based on the vectorial diffraction integral and spin density theory, such newly configurable beams are tightly focused and isotropically interfered in a 4π microscopic configuration to create three polarized field components perpendicular to each other beyond the diffraction limit, thus enabling us to yield a super-resolved focal spot possessing spatial spin axis. By further willfully adjusting the amplitude factors of the reconstituent fields, the photonic spin direction can be freely tunable. The demonstrated results in this Letter may hold great potential for the spin photonics.

Theoretical and experimental study of self-reconstruction property of astigmatic Bessel beam

In this paper, the self-reconstruction property of astigmatic Bessel beam is studied experimentally and theoretically. Based on the Fresnel diffraction integral theory and Babinet principle, the general expression of the intensity distribution of astigmatic Bessel beams passing through a circular obstacle is derived. The cross-section light intensity at transmission distance of, 10, 30, and 80 mm after astigmatism of the astigmatic Bessel beam are occluded by circular obstacles. The self-reconstruction process of the light field is observed and verified by using an specially designed experimental setup. In the experiment, we choose He-Ne laser as a light source, collimate and expand the beam through a telescope system, and a zero-order astigmatic Bessel beam is generated by a beam vertically incident on the tilted axicon after the diaphragm. A circular obstacle with a radius of 0.2 mm is placed at a distance of 200 mm behind the axicon. Finally, the cross-section intensities at different distances are observed and recorded by a microscope. The experimental phenomena are in good agreement with the theoretical prediction. The results show that the reconstruction of the zero-order astigmatic Bessel beams will occur after passing through the on-axis and off-axis obstacles. And as the transmission distance increases, the outer contour size of the astigmatic Bessel beam becomes larger, and the number of central spot arrays increases, and the complete beam is gradually reconstructed. Particularly, this feature is different from the behavior of the non-diffracting Bessel beam, which maintains the light field unchanged during transmission and has a single central spot. It is expected to be applied to multi-layer multi-particle control. And a new optical property is discovered in the experiments: the reconstruction speed of the beam in the horizontal and vertical direction are not consistent in the reconstruction process, and there is a certain speed difference. Further, we add a spiral phase plate between the diaphragm and the axicon to produce a high-order astigmatic Bessel beam. And it is verified that the high-order astigmatism Bessel beam has the same self-reconstruction characteristics after being shielded by obstacles. Compared with the zero-order aperture system, the high-order beam can not only expand the operating range, but also use the orbital angular momentum carried by the beam to achieve light rotation, which makes the particle manipulation more flexible. The research proves the self-reconstruction characteristics of astigmatic Bessel beams theoretically and experimentally, and broadens the research range of astigmatic Bessel beams. The research results have practical significance and application value in the field of optical micro-manipulation.

Sub-diffraction-limit spatially structured light pattern based on polarized beam phase modulation

The sub-diffraction-limit spatially structured light patterns have attracted more and more attention for their important applications in many frontier scientific fields. The present paper aims at developing sub-diffraction-limit spatially structured beam patterns which might have great potential to improve the light performance in fields such as super resolution imagery, optical tweezer, micro/nano lithography, etc. Here, a variety of spatially structured beam patterns are obtained by the phase modulation of polarized beams and studied in detail experimentally and numerically. Firstly, a new kind of phase plate, which combines the merits of circular and vortex 2 phase plates, is proposed based on the wave front design; it is composed of two spiral-shaped phase plates with their phases changing from 0 to 2 and - to , respectively. Later, the phase plate is applied to the circularly polarized Gaussian beam modulation in a high NA system. By combining a self-made circular with a commercial vortex 2 phase plate, the designed new phase plate is implemented in the experiment. The morphology of the spatially structured light pattern, which is generated on the focal plane, is observed by a CCD camera in the experiment. The beam pattern presents a donut shape on the focal plane, while the dimension of the donut-shaped pattern becomes smaller as the imaging plane axially deviates from the focal plane. It is found that the beam patterns captured in experiment highly consist with the numerical simulation results carried out by the vectorial diffraction integral theory. It can be deduced that the spatially structured beam is capillary-shaped. Meanwhile, at the two ends of the capillary-shaped beam, the inner diameter is smaller than the diffraction limitation. Furthermore, the structured beam pattern presents a spatial voxel distribution with center and axis symmetry. Finally, the characteristics of the spatially structured beam patterns, which are generated by modulating circular, linear, radial and azimuthal polarized beams with the new designed phase plate, are analyzed and discussed in detail. It is found that for circular, linear, radial and azimuthal polarization, the full widths at half maximum (FWHMs) of the minimum dark spots in the horizontal direction are 0.31, 0.32, 0.24 and 0.36, respectively. On the optical axis, the FWHMs of the dark spots created by linearly, radially and azimuthally polarized light, are 0.8, 0.78 and 0.76 , respectively, and no axial intensity is found with circularly polarized beam incidence.

Complex amplitude reconstruction for dynamic beam quality M2 factor measurement with self-referencing interferometer wavefront sensor.

A real-time complex amplitude reconstruction method for determining the dynamic beam quality M2 factor based on a Mach-Zehnder self-referencing interferometer wavefront sensor is developed. By using the proposed complex amplitude reconstruction method, full characterization of the laser beam, including amplitude (intensity profile) and phase information, can be reconstructed from a single interference pattern with the Fourier fringe pattern analysis method in a one-shot measurement. With the reconstructed complex amplitude, the beam fields at any position z along its propagation direction can be obtained by first utilizing the diffraction integral theory. Then the beam quality M2 factor of the dynamic beam is calculated according to the specified method of the Standard ISO11146. The feasibility of the proposed method is demonstrated with the theoretical analysis and experiment, including the static and dynamic beam process. The experimental method is simple, fast, and operates without movable parts and is allowed in order to investigate the laser beam in inaccessible conditions using existing methods.

The focused vectorial fields with ultra-long depth of focus generated by the tunable complex filter

According to the diffraction integral theory of vector field and the pupil filtering method with a discrete complex amplitude, we present a tunable pupil filter to achieve and manipulate the focused vector fields with ultra-long depth of focus. The filter consists of a polarization rotator with two /2 wave plates and a discrete complex amplitude filter with six zones. Amplitude transmissions of these zones are different and increase along the radial direction. And every two adjacent transmitted zones have the opposed phases 0 and . With optimized cylindrical polarization (0 =52) and discrete amplitude, the generalized cylindrical vector field can be tightly focused into a three-dimensional (3D) flat-top field with an extended depth of focus (～10 ) by a high numerical aperture lens. For the main outermost zone and the other five inner zones, we analyze the intensity distributions of the three polarized components and the total polarized component in the focal region. We find that the axially and azimuthally polarized components are the major contributors. The outermost zone offers the central field of the focused field while the other five zones affect the side lobe more obviously. Through adjusting the included angle between the double wave plates, we can change the polarization states of the incident vector field and alter the structures of the focused fields among the 3D flat-top focused field, needle-like field, tube-like field and the other fields with intermediate form. Result obtained is superior to that of the past research for the adjustable freedom between these forms, and it reveals the dynamic relation between the evolved vectorial form of incident field and the vector structure of the focused field. Our work paves a way to achieve the controlled dynamical focused field with a long depth of focus. The needle-like field, tube-like field and the well-matched 3D flat-top focused field will meet the demand of applications in optical microscope, optical micro-manipulating, optical micromachining and so on.

Effects of turbulent medium refractive index structure constant Cn2 on the propagation characteristics of double-half hollow Gaussian beams

Based on the Rayleigh-Sommerfeld diffraction integral theory, the propagation analytical expression of double-half Gaussian hollow beams in turbulent atmosphere is derived by using the cross spectral density function, and the effects of turbulent medium refractive index structure constant Cn2 on the propagation characteristics of double half a hollow Gaussian beams are studied and the intensity distributions under the different conditions are obtained as well. The research results show that the increased Cn2 can exacerbate the near-field diffraction effects of double-half Gaussian hollow beams, which not only shortens the propagation distance of hollow beams fully evolved into the Gaussian beams, but also increases the extent of the Gaussian beam extend outward.

Line focusing characteristics of axicon illuminated by non-diffracting Bessel beam

The line focusing characteristics of axicon illuminated by non-diffracting Bessel beam was analyzed in terms of Hankel theory and diffraction integral theory, and a new kind of method to generate periodic bottle beam was put forward, i.e. Talbot effect bottle beam was generated by axicon illuminated by non-diffracting Bessel beam. Intensity distributions along the propagation path and the intensity evolution of the bottle beams in a complete period after the axicon illuminated by non-diffracting Bessel beam were simulated numerically. An optical system was designed and Bessel-like beam was obtained by light wave emitted from He-Ne laser passing through axicon after the optical system, and then an axicon was used to focus the Bessel-like beam. Periodic bottle beam was then observed after the axicon, and the spot diagram of two periodics were taken by a CCD camera. Experimental results agree well with the theoretical analysis. This result has a practical significance in multi-plane micro-manipulation and may serve as a guideline in optical micro-control by periodic bottle beam.

Auto-alignment in four-pass amplifier with wavefront correction system

In order to solve the interaction problem between the wavefront distortion correction and optical alignment in a high-power laser system, especially in a multi-pass amplification system, based on fresnel diffraction integral theory, the beam propagation model of 310 mm×310 mm aperture beam in four-pass amplifier is built by analyzing the properties of this system. The problem of excessive amount of calculation, caused by high resolution in traditional calculation, is solved by coordinate transformation method, Using this model, the differences of far-field accuracy among centroid method, circle fitting and pure phase matching filter in beam with wavefront distortion are discussed; what influence the wavefront correction has on focal spot position is analyzed. This article provides an important basis for optimizing collimation adjustment programs in four-pass amplifier.

Sending femtosecond pulses in circles: highly nonparaxial accelerating beams

We use caustic beam shaping on 100 fs pulses to experimentally generate nonparaxial accelerating beams along a 60° circular arc, moving laterally by 14 µm over a 28 µm propagation length. This is the highest degree of transverse acceleration reported to our knowledge. Using diffraction integral theory and numerical beam propagation simulations, we show that circular acceleration trajectories represent a unique class of nonparaxial diffraction-free beam profile which also preserves the femtosecond temporal structure in the vicinity of the caustic.

Two-dimensional performance of uniform irradiation with the use of a concentricity deviation lens array and partially coherent light

A method of employing a lens array (LA) and partially coherent light (PCL) is presented for uniform irradiation on targets in inertial confinement fusion. Quasi-near field spots with a profile of sharp edge and flat top can be obtained with an LA, while the spatial coherence of the incidence beam is reduced by PCL. An appropriate concentricity deviation (CD) of LA results in the superposition of beamlets at the focus dislocated by a small amount each other and the CD and PCL reduce the fringe separations, thus the interference fringes become denser and the small-scale intensity non-uniformity of intensity distribution is suppressed. Based on the generalized diffraction integral theory, the intensity distribution of a Gaussian Schell Beam (GSB), point focused by an LA and a spherical lens is investigated in detail by two-dimensional simulation. And irradiation uniformity effects are compared with coherent light and partially coherent light after the LA. Numerical results indicate that using an appropriate CD and moving the target slightly from the exact focal plane of the principal focusing, a well-irradiated laser spot and great energy efficiency can be obtained.