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 analysis of self-convergent beam width and characterization of hard-edge diffracted beams

For a laser beam diffracted by a hard-edge aperture, propagation of the beam width, defined by the second-order moment of its irradiance distribution truncated according to the self-convergent-width criterion, obeys the familiar hyperbolic law. It is demonstrated numerically that, with the self-convergent-width approach, the beam-propagation parameters for three beam types (Gaussian, Hermite-Gaussian, and flattened Gaussian) diffracted by hard-edge apertures can be determined with the second-moment-based procedure that is recommended by the present draft standard only for unapertured laser beams.     

Analytical far-field divergence angle of a truncated gaussian beam

Approximate, but accurate, analytical expressions for the far-field divergence angle of a Gaussian beam normally incident on a circular aperture are derived. A first equation is obtained based on the concept of Gaussian transform, in which the Bessel function present in the far-field diffraction integral is approximated by a Gaussian function. Refining this approach yields another simple, practical closed-form formula with such a level of accuracy that we propose that it can be used as an exact reference. All approximations hold for any combination of Gaussian beam width and aperture radius.     

Design concept for diffractive elements shaping partially coherent laser beams.

A new two-step design algorithm for the calculation of a diffractive phase element (DPE) for use with partially coherent laser beams is presented. The optical reconstruction of the DPE is modeled by the convolution of a coherent diffraction pattern and the far-field intensity distribution of a partially coherent laser beam. Numerical deconvolution is applied to derive a suitable amplitude pattern as signal input to a standard iterative Fourier transform algorithm (IFTA). Theory and numerical results are presented. Compared with a single-step IFTA design, this new approach yields nearly equal diffraction efficiencies and a relative improvement of 15% in signal reconstruction error.     

A new synthetic aperture focusing method to suppress the diffraction of ultrasound

Spatial resolution of an ultrasound image is limited by diffraction of ultrasound as it propagates along the axial direction. This paper proposes a method for reducing the diffraction spreading effect of ultrasound by using a synthetic aperture focusing (SAF) method that uses plane waves instead of spherical waves. The new method performs data acquisition and beamforming in the same manner as conventional SAF methods. The main difference is that all array elements are used on each firing to generate a plane wave, the traveling angle of which varies with the position of a receive subaperture. On reception, each scan line is formed by synthesizing RF samples acquired by relevant receive subapertures with delays to force the plane waves to meet at each imaging point. Theoretical analysis and computer simulation with infinite transmit aperture show that the proposed method is capable of suppressing the diffraction of ultrasound and especially causing the lateral beam width to remain unchanged beyond a certain depth determined by the size of a receive subaperture and the maximum traveling angle of plane waves. It is demonstrated that the proposed method is realizable using a linear array transducer. It is also shown that the lateral radiation pattern produced by the proposed method has smaller beam width than that using conventional SAF methods in the region of interest because it suppresses the diffraction of ultrasound.     

Focal shift of focused truncated Lorentz-Gauss beam

Based on the Collins diffraction integral formula and the complex Gaussian expansion of the aperture function, an analytical expression for a Lorentz-Gauss beam focused by an optical system with a thin lens and a circular aperture has been derived. The focal shift of the focused truncated Lorentz-Gauss beam is investigated with numerical examples, and the dependence of the focal shift on the different parameters of the focused truncated Lorentz-Gauss beam is discussed in detail. This research is useful to the applications of highly divergent laser beams.     

An interference method for measuring parallelism of miniature optical components

A method for measuring parallelism of transparent optical components with small aperture size is described. It uses a Haidinger-type laser interferometer adapted for the measurement of optical components with millimeter and sub-millimeter aperture size. The method is based on the measurement of the optical thickness variation when the plate under test is translated across a focused laser beam. Measurement results for optical parts with 0.8 mm–10 mm aperture size are presented.     

Laser-near-field-based Atomic Lens: Quantum Wave Optics Consideration

Abstract

This paper examines the focusing of an atomic beam by the near field of laser radiation diffracted by a small aperture. The wave properties of the atom are taken into consideration within the framework of an approximate Schrödinger equation. The possibility is demonstrated of focusing an atomic beam into a spot some 3 nm across, the atomic and laser beam characteristics being quite acceptable. The main contribution to the spot size is shown to be from spherical aberrations and diffraction effects, the chromatic aberrations of the atomic beam with a 5% velocity distribution broadening having a lesser effect.     

A novel technique to measure laser beam spot sizes

In this letter we report a novel technique to measure small laser beam spot sizes. We use the open aperture z-scan technique as a tool to measure the laser beam spot size. This technique measures small spot sizes with accuracy better than 10%.     

Measuring the focal length of optical systems by grating shearing interferometry

Using grating shearing interferometry, a new and simple technique to measure the effective focal length of optical systems is described. The diffraction pattern of a phase grating positioned at the focal point of the lens under test is evaluated for this purpose. The relative lateral shift between the undiffracted zero order and the diffracted first orders caused by the grating is measured. By utilizing knowledge of the wavelength of light, the grating period, and the diameter of an aperture stop placed in front of the test lens, we can determine the effective focal length of the test lens. Results of measurements are presented and compared with calculated values. The dependence of the focal length on the wavelength of the light is shown by using two laser sources of different wavelengths. An analysis of the measurement accuracy is given.     

Hybrid microdiffractive-microrefractive lens with a continuous relief fabricated by use of focused-ion-beam milling for single-mode fiber coupling

The design, microfabrication, and testing of a hybrid microdiffractive-microrefractive lens with a continuous relief that is used with a laser diode for monomode fiber coupling is discussed. The hybrid microlens with a diameter as small as 65 mum and a numerical aperture of 0.25 is fabricated directly onto a spherical surface by use of focused-ion-beam milling technology. A focused Ga(+) ion beam is used to mill the continuous relief microstructure at an acceleration voltage of 50 kV. From the test results a coupling efficiency of 71% (-1.49 dB) was achieved at room temperature. A diffraction efficiency of the main diffractive order, -1, was measured to as high as 90.5% at a wavelength of 635 nm. This indicates that the hybrid microlens was suitably designed within the application requirements and satisfactorily manufactured with focused-ion-beam milling.     

Heterodyne wavelength meter for continuous-wave lasers

A wavelength meter based on a heterodyne interferometer is presented. A single-wavelength test laser beam is modulated to two orthogonal linearly polarized components with different frequencies by a pair of acousto-optic modulators. Then the modulated laser beam and a two-wavelength laser beam are sent to a heterodyne interferometer in a common path. The ratio of two laser interference phase shifts in the heterodyne interferometer is equal to the ratio of their wavelengths. The heterodyne technique measures the heterodyne interference phase but not the interference intensity, which means that it could measure a light source whose intensity is not stable. The heterodyne interference signal is an alternating signal that can easily magnify and process the circuit that makes up the heterodyne wavelength meter and could be used to measure the low-intensity light source even when there are environmental disturbances. A tunable diode laser wavelength range of 630-637 nm has been measured to an accuracy of 5 parts in 10(7).     

Effect of quadratic radial variation of phase on single slit diffraction of Laguerre–Gauss vortex beams

Effect of quadratic radial phase variation in the plane of the aperture on Fraunhofer diffraction of Laguerre–Gauss vortex beams by a slit is studied experimentally. For slit positions near the incident beam waist, its effect is to shear the diffraction pattern relative to that at the waist. The magnitude and sense of shear depend on the topological charge and slit location relative to the incident beam waist. For slit positions far from the waist, the diffraction pattern evolves to be significantly different and is dominated by two strong peaks. A closed form analytical expression for the diffraction pattern is presented, which reproduces experimental results quite well for all slit positions.     

Low Frequency Vibrating Optical System for Detecting Objects Buried in Turbid Media: Preliminary Results

Preliminary results of an in-plane vibrating system to image objects buried in turbid media are presented. The incident optical beam is vibrated in a periodic back-and-forth motion at low frequency and small constant amplitude in a plane perpendicular to the direction of the beam. The detection is performed in the AC mode, blocking the DC component. The system shows a dramatic increase in the AC signal whenever the target boundary intersects with the reference line between the incident laser beam and a photodiode after a small aperture. The system was capable to render visible 2?mm width objects buried at depths up to 3?cm from the front surface of a 1% intralipid sample.     

Micropatterning: Ultrafast Large‐Area Micropattern Generation in Nonabsorbing Polymer Thin Films by Pulsed Laser Diffraction (Small 6/2011)

An ultrafast, parallel, and beyond‐the‐master micropatterning technique for ultrathin (30?400 nm) nonabsorbing polymer films by diffraction of laser light through a 2D periodic aperture is reported. The redistribution of laser energy absorbed by the substrate causes self‐organization of polymer thin films in the form of wrinklelike surface relief structures caused by localized melting and freezing of the thin film. Unlike conventional laser ablation and laser writing processes, low laser fluence is employed to only passively swell the polymer as a pre‐ablative process without loss of material, and without absorption/reaction with incident radiation. Self‐organization in the thin polymer film, aided by the diffraction pattern, produces microstructures made up of thin raised lines. These regular microstructures have far more complex morphologies than the mask geometry and very narrow line widths that can be an order of magnitude smaller than the openings in the mask. The microstructure morphology is easily modulated by changing the film thickness, aperture size, and geometry, and by changing the diffraction pattern.     

Ultrafast large-area micropattern generation in nonabsorbing polymer thin films by pulsed laser diffraction

An ultrafast, parallel, and beyond-the-master micropatterning technique for ultrathin (30-400 nm) nonabsorbing polymer films by diffraction of laser light through a 2D periodic aperture is reported. The redistribution of laser energy absorbed by the substrate causes self-organization of polymer thin films in the form of wrinklelike surface relief structures caused by localized melting and freezing of the thin film. Unlike conventional laser ablation and laser writing processes, low laser fluence is employed to only passively swell the polymer as a pre-ablative process without loss of material, and without absorption/reaction with incident radiation. Self-organization in the thin polymer film, aided by the diffraction pattern, produces microstructures made up of thin raised lines. These regular microstructures have far more complex morphologies than the mask geometry and very narrow line widths that can be an order of magnitude smaller than the openings in the mask. The microstructure morphology is easily modulated by changing the film thickness, aperture size, and geometry, and by changing the diffraction pattern.     

Evaluation of the radiation pattern of a split aperture linear phased array for high frequency imaging

Developing transducer arrays for high frequency medical imaging is complicated because of the extremely small size and spacing of the array elements. For example, a 50 MHz linear phased array requires a center-to-center spacing of only 15 mum (one-half wavelength in water) to avoid the formation of grating lobes in the radiation pattern of the array. Fabricating an array with these dimensions is difficult using conventional technology. A split aperture design that permits much larger element spacing (3 to 4 times) while avoiding the formation of grating lobes is described. The 3-D radiation pattern of a 1.9x1.4 mm, 50-MHz split aperture linear phased array with 33 transmit elements and 33 receive elements has been evaluated theoretically. The azimuthal beam width is 90 mum at a distance of 4.0 mm. Grating lobes are suppressed by at least 60 dB at distances >4.0 mm (~f/2). The elevation beam width is 220 mum at 4.0 mm, and a useful depth of field over the axial range from 4 to 10 mm is obtained.     

An intuitive model for on-axis pulse evolution of ultrashort pulsed Gaussian beams diffracted from a circular aperture

The diffraction of ultrashort pulsed Gaussian beams from a circular aperture is studied by means of Fresnel diffraction integral and Fourier transform method. A uniform analytical expression is derived for temporal pulse form of ultrashort pulsed Gaussian beams in two cases, i.e. with constant beam waist and with constant diffraction length. It is shown that the on-axis pulse can be formulated as a superposition of an unapertured pulse and an aperture-induced pulse. The superposition of these two pulses leads to an enhanced pulse intensity for small truncation parameters at certain distances in the near field. Our results may find applications in high-intensity laser waveform control.     

Reflective diffractive beam splitter for laser interferometers

The first realization of a reflective 50/50 beam splitter based on a dielectric diffraction grating suitable for high-power laser interferometers is reported. The beam splitter is designed to operate at a wavelength of 1064 nm and in s polarization. To minimize the performance degradation of the device that is due to fabrication fluctuations, during the design process special attention was paid to achieve high fabrication tolerances especially of groove width and depth. Applying this beam splitter to high-power laser interferometers, such as future gravitational wave detectors, will avoid critical thermal lensing effects and allow for the free choice of substrate materials.     

Error sources and their impact on the performance of dual-wedge beam steering systems

This paper presents analytical and numerical results that elucidate the impact of error sources on the performance of dual-wedge beam steering systems. Different types of error sources are considered. Specifically, we investigate optical distortions in the pattern scanned out by a single ray through a pair of rotatable wedge elements with slightly different parameters. Case examples are given to reveal the difference between the distorted patterns and the patterns produced by a pair of perfectly equal wedge elements. Furthermore, nonparaxial ray tracing is performed to investigate the impact of assembly errors on the accuracy of steering a laser beam to a remote target. We found that a misalignment in a bearing axis of rotation with respect to the system optical axis will result in a change of beam deflection off-axis that gives rise to a severe decrease of pointing accuracy to a level well below the level that a tilted wedge prism may attain.