Fourier Optics
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The Green-Function Transform and Wave Propagation
1 The Green-function transform and wave propagation Colin J. R. Sheppard,1 S. S. Kou2 and J. Lin2 1Department of Nanophysics, Istituto Italiano di Tecnologia, Genova 16163, Italy 2School of Physics, University of Melbourne, Victoria 3010, Australia *Corresponding author: [email protected] PACS: 0230Nw, 4120Jb, 4225Bs, 4225Fx, 4230Kq, 4320Bi Abstract: Fourier methods well known in signal processing are applied to three- dimensional wave propagation problems. The Fourier transform of the Green function, when written explicitly in terms of a real-valued spatial frequency, consists of homogeneous and inhomogeneous components. Both parts are necessary to result in a pure out-going wave that satisfies causality. The homogeneous component consists only of propagating waves, but the inhomogeneous component contains both evanescent and propagating terms. Thus we make a distinction between inhomogenous waves and evanescent waves. The evanescent component is completely contained in the region of the inhomogeneous component outside the k-space sphere. Further, propagating waves in the Weyl expansion contain both homogeneous and inhomogeneous components. The connection between the Whittaker and Weyl expansions is discussed. A list of relevant spherically symmetric Fourier transforms is given. 2 1. Introduction In an impressive recent paper, Schmalz et al. presented a rigorous derivation of the general Green function of the Helmholtz equation based on three-dimensional (3D) Fourier transformation, and then found a unique solution for the case of a source (Schmalz, Schmalz et al. 2010). Their approach is based on the use of generalized functions and the causal nature of the out-going Green function. Actually, the basic principle of their method was described many years ago by Dirac (Dirac 1981), but has not been widely adopted. -
Optimization of the Input Fundamental Beam by Using a Spatial Filter Consisting of Two Apertures
Journal of the Korean Physical Society, Vol. 56, No. 1, January 2010, pp. 325∼328 Optimization of the Input Fundamental Beam by Using a Spatial Filter Consisting of Two Apertures Bonghoon Kang∗ Department of Visual Optics, Far East University, Chungbuk Eumseong-gun 369-700 Gi-Tae Joo Materials Science and Technology Division, Korea Institute of Science and Technology, Seoul 136-791 Bum Ku Rhee Department of Physics, Sogang University, Seoul 121-742 (Received 17 August 2009) The primary importance of the power efficiency of second-harmonic generation (SHG) is deter- mined by using the spatial shape of the fundamental input beam, its confocal parameter. The improvement of the fundamental input beam by using a spatial filter consisting of two apertures. The beam quality factor M2 of the improved fundamental beam was about 2.3. PACS numbers: 42.62.-b, 42.79.Nv, 42.25.Bs Keywords: Second-harmonic generation (SHG), Spatial filter, Beam quality factor, Confocal parameter DOI: 10.3938/jkps.56.325 I. INTRODUCTION Beam shaping requires a specific input beam. In order for the shaping to create an undistorted flat top beam or Gaussian-like beam, the input beam must be Gaus- Second harmonic generation (SHG) is widely used in sian. If the input beam does not have the proper profile, nonlinear optical spectroscopy. The optimization of the then measurements will tell the user that adjustments power efficiency of SHG is of primary importance in a to the source beam must be made before attempting to number of applications, especially when continuous wave perform the laser beam shaping. -
Precise and Fast Spatial-Frequency Analysis Using the Iterative Local Fourier Transform
Vol. 24, No. 19 | 19 Sep 2016 | OPTICS EXPRESS 22110 Precise and fast spatial-frequency analysis using the iterative local Fourier transform 1 2 2,* SUKMOCK LEE, HEEJOO CHOI, AND DAE WOOK KIM 1Department of Physics, Inha University, Incheon, 402-751, South Korea 2College of Optical Sciences, University of Arizona, 1630 East University Boulevard, Tucson, Arizona 85721, USA *[email protected] Abstract: The use of the discrete Fourier transform has decreased since the introduction of the fast Fourier transform (fFT), which is a numerically efficient computing process. This paper presents the iterative local Fourier transform (ilFT), a set of new processing algorithms that iteratively apply the discrete Fourier transform within a local and optimal frequency domain. The new technique achieves 210 times higher frequency resolution than the fFT within a comparable computation time. The method’s superb computing efficiency, high resolution, spectrum zoom-in capability, and overall performance are evaluated and compared to other advanced high-resolution Fourier transform techniques, such as the fFT combined with several fitting methods. The effectiveness of the ilFT is demonstrated through the data analysis of a set of Talbot self-images (1280 × 1024 pixels) obtained with an experimental setup using grating in a diverging beam produced by a coherent point source. ©2016 Optical Society of America OCIS codes: (300.6300) Spectroscopy, Fourier transforms; (070.4790) Spectrum analysis; (070.6760) Talbot and self-imaging effects. References and links 1. L. M. Sanchez-Brea, F. J. Torcal-Milla, J. M. Herrera-Fernandez, T. Morlanes, and E. Bernabeu, “Self-imaging technique for beam collimation,” Opt. Lett. 39(19), 5764–5767 (2014). -
Diffraction Fourier Optics
Lab 3: Diffraction & Fourier Optics This week in lab, we will continue our study of wave optics by looking at diffraction and Fourier optics. As you may recall from Lab 1, the Fourier transform gives us a way to go back and forth between time domain and frequency domain. Here we will explore how Fourier transforms are useful in optics. Learning Objectives: In this lab, students will: • Learn how to understand diffraction by extending the concept of interference (Huygens' principle) • Learn how spatial Fourier transforms arise in optical systems: o Lenses o Diffraction gratings. • Learn how to change an image by manipulating its Fourier transform. Huygen’s Principle Think back to the Pre-Lab assignment. You played around with the wave simulator (http://falstad.com/ripple/). Here you saw that we can create waves by arranging point sources in a certain way. It's done in technological applications like antenna arrays where you want to create a radio wave going in a certain direction. In ultrasound probes of the body an array of acoustic sources is used to create a wave that focuses at a certain point. In the simulation we used sources with the same phase, but if you vary the phase, you can also move the focused point around. In 1678, Huygens figured out that you could calculate how a wave propagates by making a cut in space and measuring the phase and amplitude everywhere on that plane. Then you place point emitters on the plane with the same amplitude and phase as you measured. The radiation from these point sources adds up to exactly the same wave you had coming in (Figure 1). -
15C Sp16 Fourier Intro
Outline 1. Ray Optics Review 1. Image formation (slide 2) 2. Special cases of plane wave or point source (slide 3) 3. Pinhole Camera (slide 4) 2. Wave Optics Review 1. Definitions and plane wave example (slides 5,6) 2. Point sources and spherical waves (slide 7,8) 3. Introduction to Huygen’s wavelets 1. Plane wave example (slide 9) 4. Transmission of a plane wave through a single slit 1. Narrow slits and Huygen’s wavelets (slides 10,11) 2. Wider slits up to the geometric optics limit (slide 12) 3. Time average intensity patterns from single slits (slide 13) 5. Transmission of a plane wave through two slits 1. Narrow slits and Huygen’s wavelets (slides 14-17) 2. Calculating the time averaged intensity pattern observed on a screen (slides 20-22) 6. Wave view of lenses (slides 24,25) 7. 4f optical system and Fourier Transforms 1. Introduction (slide 26) and reminder that the laser produces a plane wave in the z direction, so the kx and ky components of the propagation vector are 0 2. Demonstration that the field amplitude pattern g(x,y) produced by the LCD results in diffraction that creates non-zero k vector components in the x and y directions such that the amplitude of the efield in the fourier transform plane corresponds to the kx ky components of the fourier transform of g(x,y) Lec 2 :1 Ray Optics Intro Assumption: Light travels in straight lines Ray optics diagrams show the straight lines along which the light propagates Rules for a converging lens with focal length f, with the optic axis defined as the line that goes through the center of the lens perpendicular to the plane containing the lens: 1. -
Diffraction and Spatial Filtering P
Diffraction and Spatial Filtering P. Bennett PHY452 "Optics" Concepts Fraunhofer and Fresnel Diffraction; Spatial Filtering; Geometric optics (lenses, conjugate object- image points, focal plane, magnification); Fast-Fourier Transform; Analytic functions for N-slit apertures. Background Reading “Diffraction and Fourier Optics” (RiceUnivOpticsLab.pdf). This is an excellent write-up, particularly for theory. We follow it closely. Diffraction Physics, J. M. Cowley - definitive reference for general principles and elegant presentation. Optics, text by E. Hecht - detailed reference document with much practical information. “Geometric optics” (image formation with lenses) and “diffraction” - any UG text. Edmund Optics catalog - excellent technical info, with online tutorials and prices! SpatialFilter.xlt and *.doc Fresnel diffraction calculations (XL template and help) Special Equipment and Skills Alignment of optical components; CCD camera; Laser (Edmund P54-025, 5mW, = 635nm). Precautions Never allow the laser beam into your eye, either directly (“looking” at it) or indirectly (by spurious reflection from metal, glass, etc). Even with viewing screens, the focused spot is extremely bright, so be sure to attenuate the beam to a comfortable viewing level at all times. This will save your eyes and also the detector! Do note exceed 5V for the laser diode ($600). Handle the optical components (lenses, pinhole, apertures, slits) carefully: always store them in the vertical mounts or the wooden rack. Never lay them flat on the table, since they scratch, clog or deform easily. Lens-cleaning tissues and an air brush are available for gentle cleaning. Vinyl gloves are available for handling (mounting, etc), if necessary. Check for scratches by reflections of a bright light. -
Fourier Optics
Fourier optics 15-463, 15-663, 15-862 Computational Photography http://graphics.cs.cmu.edu/courses/15-463 Fall 2017, Lecture 28 Course announcements • Any questions about homework 6? • Extra office hours today, 3-5pm. • Make sure to take the three surveys: 1) faculty course evaluation 2) TA evaluation survey 3) end-of-semester class survey • Monday are project presentations - Do you prefer 3 minutes or 6 minutes per person? - Will post more details on Piazza. - Also please return cameras on Monday! Overview of today’s lecture • The scalar wave equation. • Basic waves and coherence. • The plane wave spectrum. • Fraunhofer diffraction and transmission. • Fresnel lenses. • Fraunhofer diffraction and reflection. Slide credits Some of these slides were directly adapted from: • Anat Levin (Technion). Scalar wave equation Simplifying the EM equations Scalar wave equation: • Homogeneous and source-free medium • No polarization 1 휕2 훻2 − 푢 푟, 푡 = 0 푐2 휕푡2 speed of light in medium Simplifying the EM equations Helmholtz equation: • Either assume perfectly monochromatic light at wavelength λ • Or assume different wavelengths independent of each other 훻2 + 푘2 ψ 푟 = 0 2휋푐 −푗 푡 푢 푟, 푡 = 푅푒 ψ 푟 푒 휆 what is this? ψ 푟 = 퐴 푟 푒푗휑 푟 Simplifying the EM equations Helmholtz equation: • Either assume perfectly monochromatic light at wavelength λ • Or assume different wavelengths independent of each other 훻2 + 푘2 ψ 푟 = 0 Wave is a sinusoid at frequency 2휋/휆: 2휋푐 −푗 푡 푢 푟, 푡 = 푅푒 ψ 푟 푒 휆 ψ 푟 = 퐴 푟 푒푗휑 푟 what is this? Simplifying the EM equations Helmholtz equation: -
The Representation of Complex Images in Spatial Frequency Domains of Primary Visual Cortex
9310 • The Journal of Neuroscience, August 29, 2007 • 27(35):9310–9318 Behavioral/Systems/Cognitive The Representation of Complex Images in Spatial Frequency Domains of Primary Visual Cortex Jing X. Zhang,1 Ari Rosenberg,2 Atul K. Mallik,3 T. Robert Husson,2 and Naoum P. Issa4 1Department of Biomedical Engineering, Illinois Institute of Technology, Chicago, Illinois 60616, and 2Committee on Computational Neuroscience, 3Committee on Neurobiology, and 4Department of Neurobiology, University of Chicago, Chicago, Illinois 60637 The organization of cat primary visual cortex has been well mapped using simple stimuli such as sinusoidal gratings, revealing superim- posed maps of orientation and spatial frequency preferences. However, it is not yet understood how complex images are represented across these maps. In this study, we ask whether a linear filter model can explain how cortical spatial frequency domains are activated by complex images. The model assumes that the response to a stimulus at any point on the cortical surface can be predicted by its individual orientation, spatial frequency, and temporal frequency tuning curves. To test this model, we imaged the pattern of activity within cat area 17 in response to stimuli composed of multiple spatial frequencies. Consistent with the predictions of the model, the stimuli activated low and high spatial frequency domains differently: at low stimulus drift speeds, both domains were strongly activated, but activity fell off in high spatial frequency domains as drift speed increased. To determine whether the filter model quantitatively predicted the activity patterns, we measured the spatiotemporal tuning properties of the functional domains in vivo and calculated expected response ampli- tudes from the model. -
Vision: from Eye to Brain (Chap 3, Part B)
Vision: From Eye to Brain (Chap 3, Part B) Lecture 7 Jonathan Pillow Sensation & Perception (PSY 345 / NEU 325) Princeton University, Spring 2015 1 more “channels”: spatial frequency channels spatial frequency: the number of cycles of a grating per unit of visual angle (usually specified in degrees) • think of it as: # of bars per unit length low frequency intermediate high frequency 2 Why sine gratings? • Provide useful decomposition of images Technical term: Fourier decomposition 3 Fourier decomposition • mathematical decomposition of an image (or sound) into sine waves. reconstruction: “image” 1 sine wave 2 sine waves 3 sine waves 4 sine waves 4 “Fourier Decomposition” theory of V1 claim: role of V1 is to do “Fourier decomposition”, i.e., break images down into a sum of sine waves • Summation of two spatial sine waves • any pattern can be broken down into a sum of sine waves 5 Fourier decomposition • mathematical decomposition of an image (or sound) into sine waves. Original image High Frequencies Low Frequencies 6 original low medium high 7 Retinal Ganglion Cells: tuned to spatial frequency Response of a ganglion cell to sine gratings of different frequencies 8 The contrast sensitivity function Human contrast sensitivity illustration of this sensitivity 9 Image Illustrating Spatial Frequency Channels 10 Image Illustrating Spatial Frequency Channels 11 If it is hard to tell who this famous person is, try squinting or defocusing “Lincoln illusion” Harmon & Jules 1973 12 “Gala Contemplating the Mediterranean Sea, which at 30 meters becomes -
Visual Performance Theoretical Limit of Resolution
Visual Performance Aspects Conditions • Resolution Limit • Illumination • Pattern Detection • Monocular vs. Binocular • Pattern Recognition • Distance • Contrast Level • On-axis vs. Off-axis • Color • Single or multiple targets • Temporal Response • Literacy & Verbal ability Theoretical Limit of Resolution θ Rayleigh Criterion 1.22λ θ = radians D For λ = 587.6 nm D ranges from 2 – 8 mm 0.090 ≤θ≤0.358 mrad 0.3 ≤θ≤1.23 minutes of arc 1 Rayleigh Criteria 1 1.6 mm Pupil 0.75 Contrast 14% Airy #1 0.5 Airy #2 Sum Relative Irradiance 0.25 Photorecptor Photorecptor Photorecptor 0 -15 -10 -5 0 5 10 15 20 25 Position (microns) Visual Acuity Visual Acuity is a measure of the smallest detail that can be resolved by the visual system. There are different types of acuity measures. Point Acuity – “Binary Star” test – typically 1 arcmin resolution Vernier Acuity – Two lines slightly offset from each other. Finds smallest detectable offset – typically 10 seconds of arc θ 2 Visual Acuity Grating Acuity – Sinusoidal or Square wave gratings are used to determine the smallest separation between peaks that can be resolved. Typically 2 arcmin. θ Visual Acuity Letter Acuity – Different Letters or Symbols need to be recognized Typically 5 arcmin. E θ F P T O Z L P E D ETDRS Landolt C Tumbling Es Lea 3 Visual Acuity & Pupil Size Visual Acuity Charts 5’ N Visual Acuity Charts are designed so the 20/20 line subtends 5 arcmin. 20/40 subtends 10 arcmin 20/10 subtends 2.5 arcmin 4 Stereo Acuity Given one object slightly closer than the other, find the smallest separation that is resolvable. -
Psychophysical Aspects of Contrast Sensitivity*
S Afr Optom 2013 72(2) 76-85 Psychophysical aspects of contrast sensitivity* Anusha Y Sukhaa and Alan Rubinb a, b Department of Optometry, University of Johannesburg, PO Box 524, Auckland Park, 2006 South Africa a<[email protected]> b<[email protected]> Received 23 November 2012; revised version accepted 9 June 2013 Abstract demonstrate stereo-pair representation of contrast visual acuities in the context of diabetic eyes. This paper reviews the psychophysical aspects The doctoral research of the first author (AYS) of contrast sensitivity which concerns components that applies similar idea to understanding both of visual stimuli and the behavioural responses inter- and intra-ocular variation of contrast visual and methods used in contrast sensitivity testing. acuities. Some discussion is included of the different types of contrast sensitivity charts available as well Key words: contrast, contrast sensitivity, as a brief background on the different types of contrast visual acuity, vision science, vision graphical representations of contrast sensitivity psychophysics, stereo-pair scatter plots, and contrast visual acuities. Two illustrations also multivariate statistics Introduction (namely spatial frequency, contrast, spatial phase, and orientation) by means of Fourier analysis. Fourier Psychophysics is a scientific discipline designed analysis is an analytical method that calculates to measure internal sensory and perceptual responses simple sine-wave components whose linear sum to external stimuli1. Sensory stimuli and behavioural forms a given complex image2, 3. The visual stimuli responses are the defining or crucial concepts in typically used in contrast sensitivity testing consist of determining contrast thresholds (for example, visual sine-wave or square-wave gratings whose luminance stimuli are used in chart-based contrast sensitivity perpendicular to the bars is modulated in sinusoidal measurements). -
Laser Beam “Clean-Up” with Spatial Filter Abstract
Laser Beam “Clean-Up” with Spatial Filter Abstract To obtain good beam quality is important for many laser applications, and a typical experimental method to obtain good beam quality is the spatial filtering. Within a spatial filtering system, a pinhole is placed on the intermediate focal plane (i.e. the Fourier plane) to get rid of the unwanted spatial frequency components. To model such systems, one must consider the diffraction from the pinhole and the diffraction property of the laser beams, and we demonstrate the spatial filtering effect in this example. 2 Modeling Task How does the size of the pinhole, located at the intermediate focal plane, influence the output beam? ? input field - wavelength lens #1 lens #2 632.8 nm feff = 25 mm feff = 25 mm - multimode laser D = 12.5 mm D = 12.5 mm with “noise” pinhole circular opening Fourier plane 3 Spatial Filter with 7.5µm-Diameter Opening A relatively large pinhole does not filter out much noise on the Fourier plane. 7.5 µm behind pinhole in front of pinhole input field pinhole 7.5 µm diameter lens #1 lens #2 4 Spatial Filter with 7.5µm-Diameter Opening 7.5 µm behind pinhole in front of pinhole input field output field pinhole 7.5 µm diameter lens #1 lens #2 5 Spatial Filter with 5.0µm-Diameter Opening 5.0 µm behind pinhole in front of pinhole input field output field pinhole 5.0 µm diameter lens #1 lens #2 6 Spatial Filter with 2.5µm-Diameter Opening 2.5 µm behind pinhole in front of pinhole input field output field pinhole 2.5 µm diameter lens #1 lens #2 7 Output Beam Profile and Power Comparison lens #1 lens #2 pinhole varying diameter Spatial filter in the Fourier plane helps reduce the higher-frequency noise, therefore leads to better output beam profiles.