Spot size, depth-of-focus, and diffraction ring intensity formulas for truncated Gaussian beams

Simple polynomial formulas to calculate the FWHM and full width at 1/e2 intensity diffraction spot size and the depth of focus at a Strehl ratio of 0.8 and 0.5 as a function of a Gaussian beam truncation ratio and a system f-number are presented. Formulas are obtained by use of the numerical integration of a Huygens-Fresnel diffraction integral and can be used to calculate the number of resolvable spots, the modulation transfer function, and the defocus tolerance of optical systems that employ laser beams. I also derived analytical formulas for the diffraction ring intensity as a function of the Gaussian beam truncation ratio and the system f-number. Such formulas can be used to estimate the diffraction-limited contrast of display and imaging systems.     

Subwavelength focusing of laser light by microoptics

Near-field scanning optical microscope (NSOM) measurements revealed that a linearly polarized Gaussian beam of wavelength λ?=?532?nm focused with a binary zone plate (ZP) of focal length λ, radius 7?μm, and groove depth 510?nm, fabricated in hydrogen silsesquioxane, produces a focal spot of size FWHM?=?(0.44?±?0.02)λ, with the side lobes being lower than 10% of the intensity peak in the focus. Replacing the incident 532?nm wavelength with a 633?nm wavelength resulted in a 1.8 times shorter focal length and a tighter (in terms of wavelengths) focal spot of FWHM?=?(0.40?±?0.02)λ. This value is smaller than the scalar diffraction-limited size in vacuum, FWHM?=?0.51λ. This is the smallest focal spot so far experimentally obtained for a binary phase ZP and the root-mean-square deviation of the experimental curve from a FDTD simulation is 5%. We show that the metallic pyramid-shaped cantilever with a 100-nm-hole in the tip that is used in the NSOM is only able to detect the transverse electric field component. The FDTD simulation showed such a cantilever to be over 3 times more sensitive to the transverse electric field component than to the longitudinal one. Using the Richards–Wolf (RW) formulae, the near-focus intensity distribution can be calculated with 6% error for focal lengths larger than 4λ. It is usually assumed that the Debye theory and the RW formulae are only valid for focal lengths much larger than the incident wavelength. By FDTD simulation, we showed that when illuminating the ZP by a radially (rather than linearly) polarized beam, a decrease in the focal spot transverse size did not result in a substantially reduced total volume of focus (4%).     

Generation of a Bessel beam of variable spot size

We report the generation of a zero-order Bessel beam of continuously variable spot size using a simple optical setup. We have used a pair of metal axicon mirrors to generate a hollow beam of variable dark diameter. This beam was subsequently focused by a convex lens to get a Bessel beam of variable spot size. We also studied the effect of a hollow-beam ring width on nondiffracting propagation range of the generated beam.     

New scanning technique for the optical vortex microscope

In the optical vortex microscopy the focused Gaussian beam with optical vortex scans a sample. An optical vortex can be introduced into a laser beam with the use of a special optical element--a vortex lens. When moving the vortex lens, the optical vortex changes its position inside the spot formed by a focused laser beam. This effect can be used as a new precise scanning technique. In this paper, we study the optical vortex behavior at the sample plane. We also estimate if the new scanning technique results in observable effects that could be used for a phase object detection.     

Multiframe superresolution of binary images

We describe a new algorithm for superresolving a binary object from multiple undersampled low-resolution (LR) images that are degraded by diffraction-limited optical blur, detector blur, and additive white Gaussian noise. Two-dimensional distributed data detection (2D4) is an iterative algorithm that employs a message-passing technique for estimating the object pixel likelihoods. We present a novel non-training-based complexity-reduction technique that makes the algorithm suitable even for channels with support size as large as 5 x 5 object pixels. We compare the performance and computational complexity of 2D4 with that of iterative backprojection (IBP). In an imaging system with an optical blur spot matched to the object pixel size, 2 x 2 undersampled measurement pixels, and four LR images, the reconstruction error measured in terms of the number of pixel mismatches for 2D4 is 300 times smaller than that for IBP at a signal-to-noise ratio of 38 dB.     

Modified Fresnel zone plates that produce sharp Gaussian focal spots

A modified Fresnel zone plate that can produce an approximate Gaussian focal spot is proposed for the focusing and imaging of soft x rays and extreme ultraviolet radiation. The selection conditions for the positions and the widths of the concentric open rings are analytically presented. The focal spot size can be much smaller than the width of the narrowest open ring, and the sidelobes and the higher orders can be effectively suppressed. Through numerical experiments, we confirm that a Gaussian focal spot with a beam width of 7.7 nm can be produced by a modified Fresnel zone plate with a minimum structure size of 30 nm.     

Apodized multilevel diffractive lenses that produce desired diffraction-limited focal spots

An apodized, multilevel diffractive lens that can produce a desired diffraction-limited focal spot is proposed for many applications, such as focusing, imaging, optical storage, and optical trapping. The three key points for the design are the innovative idea of complex conjugate subzones, the use of Babinet's principle, and the equivalent-pupil (or aperture) function theory of diffractive focusing elements composed of concentric transparent rings. As a concrete example, we numerically design a mixed multilevel diffractive lens (the highest phase level is 8) to produce a diffraction-limited Gaussian focal spot. Some related problems, such as the validity range and the combination with high-numerical-aperture refractive lenses, are also discussed.     

Effective beam parameters and the turbulent beam waist for convergent Gaussian beams

Expressions are developed for the location and the size of the beam waist for a convergent Gaussian beam in statistically homogeneous and isotropic atmospheric turbulence. Subsidiary expressions are presented that lead to the maximum distance from the transmitter at which the beam waist can be located under given optical turbulence conditions and the optimal initial radius of curvature required for placing the beam waist at a desired location. The free-space beam radius W of a Gaussian beam satisfies the relationship ?W/?z = - W/R, where z represents the path length and R is the phase-front radius of curvature at z. By enforcing this relation on the effective beam spot size in turbulence W(e), we can define an effective radius of curvature R(e). In addition to specifying the beam waist, R(e) leads to a pair of effective beam parameters θ(e) and Λ(e) that provide a natural extension to the complex amplitude plane. Within this context, general propagation characteristics may be described, including the coherence properties of a Gaussian beam in both weak and strong optical turbulence.     

Focal shift of a focused partially coherent Laguerre-Gaussian beam of all orders

Abstract

We explore the focusing properties of a partially coherent Laguerre–Gaussian (LG) beam of all orders, particularly the focal shift (i.e. the shift of the actual focal plane away from the geometrical focal plane). We derive the analytical expressions for the average intensity and the effective beam width of a focused partially coherent LG beam, and we adopt the minimum effective beam width instead of the conventional maximum on-axis intensity to determine the actual focal plane. It is found that the focused beam shape, minimum effective beam width and the focal shift of a focused partially coherent LG beam are determined by its initial coherence width, radial mode order and azimuthal mode order (i.e. topological charge) together. Our results may be useful for optical trapping and micro-fabrication, where precise focal position and prescribed beam shape are required.     

Influence of optical aberrations on laser-induced plasma formation in water and their consequences for intraocular photodisruption

The influence of spherical aberrations on laser-induced plasma formation in water by 6-ns Nd:YAG laser pulses was investigated for focusing angles that are used in intraocular microsurgery. Waveform distortions of 5.5lambda and 18.5lambda between the optical axis and the 1/e(2) irradiance values of the laser beam were introduced by replacement of laser achromats in the delivery system by planoconvex lenses. Aberrations of 18.5lambda increased the energy threshold for plasma formation by a factor of 8.5 compared with the optimized system. The actual irradiance threshold for optical breakdown was determined from the threshold energy in the optimized system and the spot size measured with a knife-edge technique. For aberrations of 18.5lambda the irradiance threshold was 48 times larger than the actual threshold when it was calculated by use of the diffraction-limited spot size but was 35 times smaller when it was calculated by use of the measured spot size. The latter discrepancy is probably due to hot spots in the focal region of the aberrated laser beam. Hence the determination of the optical-breakdown threshold in the presence of aberrations leads to highly erroneous results. In the presence of aberrations the plasmas are as much as 3 times longer and the transmitted energy is 17-20 times higher than without aberrations. Aberrations can thus strongly compromise the precision and the safety of intraocular microsurgery. They can further account for a major part of the differences in the breakdown-threshold and the plasma-transmission values reported in previous investigations.     

Fault localization and analysis in semiconductor devices with optical-feedback infrared confocal microscopy

We report on a cost-effective optical setup for characterizing light-emitting semiconductor devices with optical-feedback confocal infrared microscopy and optical beam-induced resistance change. We utilize the focused beam from an infrared laser diode to induce local thermal resistance changes across the surface of a biased integrated circuit (IC) sample. Variations in the multiple current paths are mapped by scanning the IC across the focused beam. The high-contrast current maps allow accurate differentiation of the functional and defective sites, or the isolation of the surface-emitting p-i-n devices in the IC. Optical beam-induced current (OBIC) is not generated since the incident beam energy is lower than the bandgap energy of the p-i-n device. Inhomogeneous current distributions in the IC become apparent without the strong OBIC background. They are located at a diffraction-limited resolution by referencing the current maps against the confocal reflectance image that is simultaneously acquired via optical-feedback detection. Our technique permits the accurate identification of metal and semiconductor sites as well as the classification of different metallic structures according to thickness, composition, or spatial inhomogeneity.     

远距离高精度多普勒位移测量

通过泛函、激光散斑理论和随机过程各态遍历的研究,导出激光多普勒信号强度与聚焦光斑直径、接收透镜通光口径、光电接收器响应等参数间的关系,在此基础上设计出一种适用于远距离处面内位移测量的光路。此光路将高斯光束束腰聚焦在被测体上,实现最小聚焦光斑和平面波叠加,此外采用大口径透镜接收散射光,用响应度高的光电接收器转换光电信号等措施获得高强度高信噪比的测量信号。将此光路用于 100m 处面内位移(49.70mm)测量,其精度可达 2%。此设计方法能用于振动或地震波的高精度检测。     

Variational analysis of z scan of thick medium with an elliptic Gaussian beam

By variational approach, we analyze the characteristics of beam propagation through a cubic optical nonlinear medium using a laser beam that has a transverse elliptic Gaussian profile. The analytic solution to the normalized transmittance at the center of the far field as a function of medium position and the beam characteristics is obtained and compared with the numerical simulation, which is realized by a combination of algorithms. We also analyze the peak-valley transmittance difference as a function of medium length, ellipticity, and a stigmatism. The relationship between peak-valley normalized transmittance difference of the z-scan trace and aperture size or the slit width are obtained. Meanwhile, the comparison of z-scan characteristics with an elliptic Gaussian beam with those using a circular symmetric Gaussian beam is made.     

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.     

Collimating properties of an axicon for the Čerenkov-radiation-type second-harmonic light from a crystal-cored fiber

The optical second-harmonic wave generated from a crystal-cored fiber from the ?erenkov-radiation-type phase matching can be collimated by an axicon and then can be focused by a spherical lens to a diffraction-limited spot. Tolerance of the optical system to meet the Maréchal criterion is analyzed; tolerable maximum errors in alignment and fabrication of the axicon lens are evaluated.     

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.     

Adaptive conversion of a high-order mode beam into a near-diffraction-limited beam

We present a new method for efficiently transforming a high-order mode beam into a nearly Gaussian beam with much higher beam quality. The method is based on modulation of phases of different lobes by stochastic parallel gradient descent algorithm and coherent addition after phase flattening. We demonstrate the method by transforming an LP11 mode into a nearly Gaussian beam. The experimental results reveal that the power in the diffraction-limited bucket in the far field is increased by more than a factor of 1.5.     

Atmospheric optical communication with a Gaussian Schell beam

We consider a wireless optical communication link in which the laser source is a Gaussian Schell beam. The effects of atmospheric turbulence strength and degree of source spatial coherence on aperture averaging and average bit error rate are examined. To accomplish this, we have derived analytic expressions for the spatial covariance of irradiance fluctuations and log-intensity variance for a Gaussian beam of any degree of coherence in the weak fluctuation regime. When spatial coherence of the transmitted source beam is reduced, intensity fluctuations (scintillations) decrease, leading to a significant reduction in the bit error rate of the optical communication link. We have also identified an enhanced aperture-averaging effect that occurs in tightly focused coherent Gaussian beams and in collimated and slightly divergent partially coherent beams. The expressions derived provide a useful design tool for selecting the optimal transmitter beam size, receiver aperture size, beam spatial coherence, transmitter focusing, etc., for the anticipated atmospheric channel conditions.     

Acoustooptic 2-D profile shaping of a Gaussian laser beam

Acoustooptic 2-D profile shaping of a Gaussian laser beam has been achieved by two plane ultrasonic waves progressing in orthogonal directions. The spot size W of the Gaussian laser beam must be considerable less than the wavelength lambda of the ultrasonic wave at the acoustooptic interaction region. The ultrasonic cell is dealt with as a Raman-Nath 2-D phase grating but serves as a 2-D beam deflector in time for the interaction scheme of interest. The wave front of the Gaussian laser beam must be almost plane in the interaction region. The profile shaping condition is 0.15 < or = (W/lambda) < or = 0.30 only when the Raman-Nath parameter dependent on the ultrasonic power has values between v = 1.0 and 2.0.     

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.