Fraunhofer Diffraction Research Articles

A simple determination approach for zero-padding of FFT method in focal spot calculation

FFT is a widely utilized method to calculate focal spot, in which zero padding is needed to adjust the sampling interval on the focal plane. This work is to apply Abbe limit of resolution to determining of the number of padded zeros when sampling interval on focal plane is given. We specifically investigate the correlation of digitized Fraunhofer diffraction and discrete Fourier transform (DFT) in a lens model. The theoretical support for zero padding on the back focal plane (BFP) is illustrated based on a hypothesis of lens with finite numerical aperture. The proposed hypothesis and the zero padding approach are verified by focal spots calculation, where lights polarized linearly and radially are used as samples.

Tropospheric Effect on the Transmissivity of the Spacecraft–Ground Tracking Station Communication Line

Algorithms ensuring high-speed and reliable data transmission under lognormal amplitude fluctuations (described by Fraunhofer diffraction) in the spacecraft–ground tracking station line for coherent and incoherent reception of signals have been considered. An advantage of the coherent reception of millimeter-range signals with a random error-correcting code has been indicated.

Fraunhofer diffraction and the state of polarization of partially coherent electromagnetic beams.

We generalize the concept of Fraunhofer diffraction to partially coherent electromagnetic beams and show how the state of polarization is affected by a circular aperture. It is illustrated that the far-zone properties of a random beam can be tuned by varying the aperture radius. We find that even an incident beam that is completely unpolarized can sometimes produce a field that is highly polarized.

Design and qualification of a recessed satellite cornercube retroreflector for ground-based attitude verification via satellite laser ranging

The design and qualification of a retroreflector specifically designed for a CubeSat (CubeL) is described. The CubeSat will be launched to space in 2019 and demonstrate the latest generation of the optical space infrared downlink system developed by the German Aerospace Center together with the industrial partner Tesat-Spacecom. The retroreflector is optimized to allow for a coarse verification of the satellites attitude control system. By analyzing the returning photon count during satellite laser ranging (SLR) when the satellite is operated in station pointing mode attitude information is obtained. To achieve this goal, the entrance face of the retroreflector is recessed by a circular tube-shaped aperture. Due to this recession, the signal reflected from the retroreflector falls off rapidly when the retroreflector is tilted away from the SLR station. From measurements of the retroreflectors far-field diffraction pattern and calculations, we believe that it should be possible to determine the orientation accuracy of the satellite to within ± 2°. The proposed method is an effective and cheap way for coarse attitude control, e.g., for satellites of mega-constellations with any existing satellite laser ranging ground station.

Optofluidic refractive index sensor based on asymmetric diffraction.

A novel optofluidic refracrtive index (RI) sensor was proposed based on asymmetric Fraunhofer diffraction. In-plane optofluidic lens, light source, slit, diffraction pattern visualization zone and optical path were integrated into the microfluidic networks to avoid the manual alignment of the optical components as well as to reduce the cost of external bulky components. Unlike the conventional RI sensor, this device visualizes the bulk refractive index change of the liquid through a diffraction image, which is readily read-out for clinical diagnosis right at the point-of-care or on-site security check. In the experiment, the device can measure a RI change of as low as ~10-5 RIU. A low noise-equivalent detection limit (NEDL) of ~10-6 refractive index unit (RIU) and high sensitivity of ~1.1 × 104/RIU were achieved. The new device is practical and suitable to be extended for high throughput applications by simultaneously reading multiple chips with an 2D-array image sensor.

Wavefront Splitting Fiber Mach–Zehnder Interferometer Modulated by the White Light Fraunhofer Diffraction

Stability, robustness, sensitivity, and reproducibility are important parameters in the interferometry. We propose a wavefront splitting fiber Mach–Zehnder interferometer (MZI) to achieve such performances. A polyethylene terephthalate (PET) film splits the wavefront of a Gaussian beam emitted from the first fiber collimator. Then, the Gaussian beam is focused into a single mode fiber by another collimator, constituting a stable and robust fiber MZI. The effective spectral width is larger than 390 nm, which is wide enough for constructing a sensitive fiber MZI. The COMSOL multiphysics simulation and the experimental results demonstrate that this MZI is based on the multi-beam interference. Thanks to the low insertion loss of the fiber MZI, two types of stable and repeatable interference fringes are obtained when the sampling accuracy of the optical spectrum analyzer is as high as 0.1 nm. It ensures that the fiber sensor based on this MZI has a high resolution. For the first type of interference fringe, the two adjacent resonant wavelengths, respectively, shift –0.78 and –0.81 nm when salinity increases one part per thousand, showing high sensitivities. The low order interference fringe has a larger sensitivity. However, its visibility may be small due to the modulation of the second type of interference fringe that has small sensitivity to the refractive index of the sample. Since the PET film has uniform thickness, the fiber MZI can ensure good reproducibility. It can be widely utilized in detecting physical, chemical, or biological quantities.

Dynamic holographic imaging of real-life scene

In this paper, a Kinect based three-dimensional real-life holographic imaging method is proposed. In this method, the three-dimensional information of a real-life scene is acquired by using a commercial Kinect device. The output depth map is converted into a point cloud using the pinhole camera model. Surface reconstruction of the point cloud is achieved by simply using directly uniform sampling method. Our proposed spatial domain Fraunhofer diffraction algorithm is used to generate dynamic hologram videos. Optical reconstruction is performed to test the validity of the proposed method. A universal image quality index is introduced to analyze the quality of holographic imaging. A depth map based triangular refinement method is used to increase the resolution of the 3D model. The final resolution of the reconstructed holographic image can be 3.0 times higher than the original one.

On the reflection and diffraction of carbon nanotube array thin film

A theory for determining the effective refraction index of carbon nanotube array thin film is developed based on Fresnel’s equation of wave normals. Here, the theory of a metallic film is used to determine the reflectivity and then to analyze the results of previous experiments on the optical properties of vertically aligned carbon nanotube arrays for different polarized incident lights. Both transverse electric-polarized lights and transverse magnetic-polarized lights are investigated, and the results agree well with the available experimental data. In a different case, when considering the weighted average of the polarized lights, the theoretical reflectivity for a non-polarized incident light is in excellent agreement with the experimental data. The refraction efficiency of a grating-patterned carbon nanotube array thin film is also investigated using the Fraunhofer diffraction theory. Based on the comparison of the theoretical and experimental results, it is concluded that the proposed theoretical model reveals the optical behavior of a carbon nanotube array thin film.

Terahertz dispersion using multi-depth phase modulation grating.

We present a multi-depth phase modulation grating (MPMG) in the terahertz range making real-time multichannel Fourier-transform spectroscopy available in a stationary manner. The calculation of the Fraunhofer diffraction field distribution and diffraction efficiency of an MPMG indicates that the zeroth-order diffraction light of an MPMG carries phase information and its diffraction intensity is modulated by the groove depth. A good agreement is found between the measurements of the 0th- and ±1st-order diffraction efficiency at 0.5 and 0.34 THz and the simulation. The frequencies of the terahertz source retrieved from the zeroth-order diffraction intensity at 0.5, 0.4, and 0.34 THz are identical to the actual frequencies.

An Optical Diffraction Method for Controlling the Solid-Phase Recrystallization and Heating of Implanted Semiconductors during Pulse Light Annealing

t—The results of the improvement of the method of laser diagnostics for investigating the heating dynamics, solid-state recrystallization, and melting of implanted semiconductors directly during a pulse light annealing procedure are presented. This technique is based on recording Fraunhofer diffraction patterns from special periodic structures; it allows investigations of structural–phase transitions in ion-doped semiconductor layers simultaneously with sample temperature measurements with a high time resolution. For this purpose, two measuring diffraction gratings were preliminarily formed on the surface of a silicon wafer: a phase grating and an amplitude grating. The solid-phase recrystallization and melting processes were studied via the kinetics of the disappearance and appearance of the diffraction maxima from the amplitude grating. The sample temperature was monitored by the deflection of the diffraction angle of the probe laser beam from the phase diffraction grating. This deflection is caused by a change in the grating period due to its thermal expansion.

Alineación Láser para Experimentación con Fotografía Digital de Speckle (DSP)

Non-contact techniques have returned to optical metrology one of the most booming branches of science, making digital speckle photography (DSP) one of the most used in the measurement of structural changes in surfaces. This technique uses a laser as a measurement system and compares the changes in the speckle pattern obtained by irradiating the surface that is subjected to deformations; however, the alignment of these systems plays an important role in the results obtained. In this work, the alignment errors of an optical system are estimated based on the analysis of the Fraunhofer diffraction pattern, as well as the statistical properties of the speckle pattern for field near and far. The results indicate that the alignment of the light beam can be determine by a fringes quantification system when passing through a pinhole and that the highly developed speckle pattern is more similar to the intensity distribution shown in the literature.

Diffraction imaging of cracks in self-assembled photonic crystals

We demonstrate a non-invasive technique, based on diffraction imaging, to remotely detect and characterize cracks in self-assembled photonic crystals. Through this technique, we measure the diffraction pattern of the cracks and determine how the orientation of the cracks varies across the sample. In particular, we characterize cracks in photonic crystals fabricated using evaporation-induced self-assembly process on a horizontal substrate. The morphological details of cracks obtained using our imaging technique match well with scanning electron microscopy images. We also develop a model for the surface of these crystals and calculate Fraunhofer diffraction patterns, using two-dimensional Fourier transform, that match well with our experimental observations.

A lotus leaf based random laser

Abstract Random lasing in a lotus leaf, with a wide tunable spectrum from planar liquid waveguide gain channels is described. The lotus leaf shows multiple scattering from the micro-papilla and nanoscale coralloid tomentum on its surface. The Fraunhofer diffraction pattern demonstrates excellent coherence and directionality for our random laser. The emission spectrum wavelength can be tuned by changing the pump position due to the random distribution of the micro/nano-scale features on the lotus leaf. Potential applications of the random laser include probing micro/nano-scale structural alterations, optical biosensors on chips, and developing a multi-color random laser.

Electron holography on Fraunhofer diffraction.

Electron holography in Fraunhofer region was realized by using an asymmetric double slit. A Fraunhofer diffraction wave from a wider slit worked as an objective wave interfered with a plane wave from a narrower slit as a reference wave under the pre-Fraunhofer condition and recorded as a hologram. Here, the pre-Fraunhofer condition means that the following conditions are simultaneously satisfied: single-slit observations are performed under the Fraunhofer condition and the double-slit observations are performed under the Fresnel condition. Amplitude and phase distributions of the Fraunhofer diffraction wave were reconstructed from the hologram by the Fourier transform reconstruction method. The reconstructed amplitude and phase images corresponded to Fraunhofer diffraction patterns; in particular, the phase steps of π at each band pattern in the phase image were confirmed. We hope that the developed Fraunhofer electron holography can be extended to a direct phase detection method in the reciprocal space.

Impact of aperture thickness on a Fraunhofer diffraction pattern

In this paper, we investigate in detail the effect of increasing aperture thickness on a Fraunhofer diffraction pattern. The scalar diffraction theory is employed to derive the solution of the far-field amplitude for a single-slit and a circular aperture in both normal and oblique incidents. Our results show that increasing the aperture thickness indisputably modifies the diffraction pattern, mainly for the wide apertures. It is proven that the maximum value of the diffracted field amplitude decreases for the non-zero values of the aperture thickness, resulting in the reduction of the fringe visibility. This study reveals that in the case of the single-guided-mode operation, the diffraction pattern is independent of the incident beam direction, and is the same for an aperture thickness larger than the attenuation length of the first evanescent mode. On the contrary, it is observed that the diffraction pattern of a wide aperture at the oblique incidence is entirely different from the popular laterally shifted normal incident diffraction profile. Therefore, we conclude that it is impossible to carefully examine the diffraction phenomena, without considering the effect of the aperture thickness.

Position Measurement-Induced Collapse: A Unified Quantum Description of Fraunhofer and Fresnel Diffractions

Position measurement-induced collapse states are shown to provide a unified quantum description of diffraction of particles passing through a single slit. These states, which we here call `quantum location states', are represented by the conventional rectangular wave function at the initial moment of position measurement. We expand this state in terms of the position eigenstates, which in turn can be represented as a linear combination of energy eigenfunctions of the problem, using the closure property. The time-evolution of the location states in the case of free particles is shown to have position probability density patterns closely resembling diffraction patterns in the Fresnel region for small times and the same in Fraunhofer region for large times. Using the quantum trajectory representations in the de Broglie-Bohm, modified de Broglie-Bohm and Floyd-Faraggi-Matone formalisms, we show that Fresnel and Fraunhofer diffractions can be described using a single expression. We also discuss how to obtain the probability density of location states for the case of particles moving in a general potential, detected at some arbitrary point. In the case of the harmonic oscillator potential, we find that they have oscillatory properties similar to that of coherent states.

Pattern recognition system based on photoanisotropic copies.

In this paper, we present a method of pattern recognition based on obtaining the photoanisotropic copies of object images. Such copies are induced on the dynamic polarization-sensitive material when the amplitude image of the object formed by a spatial light modulator is transilluminated by linearly polarized light with a wavelength actinic for the used material. In this case, the distribution of the intensities over the amplitude image is converted into the distribution of photoanisotropy over the sample of a polarization-sensitive material. In the recognition process, this copy is illuminated by parallel circularly polarized nonactinic light. The distribution of elliptical polarization occurs directly behind the photoanisotropic copy, and it is reduced to a summary ellipse in the Fraunhofer diffraction region. It has been shown that the parameters of this ellipse are related to the characteristics of the object and uniquely identify the initial object. The polarization-holographic diffraction element that we developed enables determining the polarization state of a summary ellipse to obtain all the Stokes parameters in real time. The obtained results were compared with the Stokes parameters of the recognizable object in the database. The method invariance with regard to the position, scale, and rotation of the pattern were investigated. In addition, the resolution and sensitivity of this method were determined. The polarization-sensitive materials used to obtain copies are reversible, with practically an unlimited number of recording-deleting cycles. To obtain a photoanisotropic copy of another object on the same material, the previous copy should be deleted with a pulse of circularly polarized actinic light, and then the next copy can be recorded. A laboratory model of the recognition device and the appropriate software were created, and a theoretical model was developed. Databases were obtained for the images of various objects.

Fraunhofer diffraction of a Gaussian beam by a four-sector binary grating with a half period fringes shift between adjacent sectors

The study of Fraunhofer diffraction of a Gaussian laser beam by a computer-generated, binary four-sector grating (FSG) is presented. In the four equal angular sectors of the FSG, parts of a binary rectilinear grating are embedded in such a way that, two neighboring parts are shifted by a half spatial rectilinear grating period. Analytical expressions, describing the diffracted wave field amplitude and intensity distributions in the focal plane of a converging lens, are derived. The zeroth-diffraction-order beam is a Gaussian beam whose transverse cross section has a reduced Gaussian radius. The higher-diffraction-order beam transverse intensity profiles have four bright spots divided by dark crossed lines and a central dark, non-vortex core. They can be described as cosine-Laguerre–Gaussian Kummer beams. The numerically calculated diffraction patterns, as well as the interference patterns of the diffracted beam with a slightly inclined plane wave, are supported by experimentally obtained ones. In both investigations (theoretical and experimental) the interferograms confirm the non-vortex nature of the diffraction spots.

Optical sensitivity enhancement in grating based micromechanical accelerometer by reducing non-parallelism error.

We report on optical sensitivity enhancement in a grating-based micromechanical accelerometer, which was achieved by reducing the non-parallelism error between the grating and reflected mirror. Based on the multi-slit Fraunhofer diffraction theory, an equivalent optical model is proposed in order to discuss the non-parallelism induced error that is caused by the residual stress in material and fabrication. An integrated fabrication flow with optimized quartz based and silicon based procedure is then presented to improve the parallelism between the grating and mirror, and to realize a hermetic package using silicon islands for the electrical interconnection. We experimentally characterize accelerometers' behavior by an interferometric beam detecting setup, which reveals the acceleration measurement with a scale factor improvement, noise floor decrease, and thus a bias stability enhancement from 2 mg to 0.35 mg (20 seconds interval, 1 g = 9.8 m/s2).

Vortex strength and beam propagation factor of fractional vortex beams.

Fractional vortex beams (FVBs) with non-integer topological charges attract much attention due to unique features of propagations, but different viewpoints still exist on the change of their total vortex strength. Here we have experimentally demonstrated the distribution and number of vortices contained in FVBs at the Fraunhofer diffraction region. We have verified that the jumps of total vortex strength for FVBs happen only when non-integer topological charge is before and after (but very close to) any even integer number that originates from two different mechanisms for generation and movement of vortices on focal plane. Meanwhile, we have also measured the beam propagation factor (BPF) of such FVBs and have found that their BPF values almost increase linearly in the x component (along the initial edge dislocation) and oscillate increasingly in the y component (vertical to the initial edge dislocation). Our experimental results are in good agreement with numerical results.