DOCSLIB.ORG

Geometrical Optics Lenses Principal Rays Gaussian Lens Formula: 1 1 1   So Si F F I Optical Image Object Axis Fo

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometrical Optics Lenses Principal Rays Gaussian Lens Formula: 1 1 1   So Si F F I Optical Image Object Axis Fo Phys 322 Lecture 13 Chapter 5 Geometrical Optics Lenses Principal rays Gaussian lens formula: 1 1 1 so si f F i Optical Image Object axis Fo Principal rays: 1) Rays parallel to principal axis pass through focal point Fi. 2) Rays through center of lens are not refracted. 3) Rays through Fo emerge parallel to principal axis. In this case image is real, inverted and enlarged Assumptions: Since n is function of , in reality each color has different focal point: • monochromatic light chromatic aberration. Contrast to • thin lens. mirrors: angle of incidence/reflection • rays are all “near” the principal axis not a function of (paraxial). Converging lens: examples This could be used in a camera. Big object on small film so > 2F This could be used as a projector. Small slide(object) on big F < so < 2F screen (image) This is a magnifying glass 0 < si < F Lens magnification so y Fi Image o optical axis Object f Fo yi 1 1 1 si so si f Green and blue triangles are similar: Magnification equation: yi si MT yo so Example: f=10 cm, so=15 cm T = transverse 1 1 1 30cm si = 30 cm MT 2 15cm si 10cm 15cm Thin lens equations s 1 1 1 M i T s s s f o o i f= 100 mm Real image formed formed by single lens is always inverted Longitudinal magnification 1 1 1 so si f The 3D image of the horse is distorted: • transverse magnification changes along optical axis • longitudinal magnification is not linear Longitudinal magnification: 2 Negative: a horse looking dxi f 2 M L M towards the lens forms an image dx x2 T o o that looks away from the lens dx d x x f 2 2 i 2 2 2 o i xi f / xo f / xo f / xo dxo dxo Thin lens combinations In most of applications several lenses are used Example: multiple lenses Image from lens 1 becomes object for lens 2 1 2 Fi1 Fi2 Lens 1 creates a real, inverted and enlarged image of the object. Lens 2 creates a real, inverted and reduced image of the image from lens 1. The combination gives a real, upright, enlarged image of the object. Example: multiple lenses so1 = 15 cm 1 2 Fi1 Fi2 f1 = 10 cm f2 = 5 cm si1 = 30 cm First find image from lens 1. 1 1 1 si1 = 30 cm 15cm si1 10cm Example: multiple lenses so1 = 15 cm 1 2 d = 42 cm si2 = 8.6 cm Fi1 Fi2 f1 = 10 cm f2 = 5 cm s = 30 cm i1 so2 =12 cm Now find image from lens 2. 1 1 1 si2 = 8.6 cm 12cm si2 5cm Example: multiple lenses, magnification so1 = 15 cm 1 2 d = 42 cm si2 = 8.6 cm Fi1 Fi2 f1 = 10 cm s f2 = 5 cm M i s = 30 cm T i1 so2=12 cm so 30cm Lens 1: M 2 Total magnification: T1 15cm M M M 1.44 8.6cm T T1 T 2 Lens 2: MT 2 .72 12cm +: image is not inverted Two lens equation so1 1 d 2 si2 Fi1 Fi2 f1 f2 s i1 so2 f2d f2so1 f1 /so1 f1 For combination of two thin lenses: si2 d f2 so1 f1 /so1 f1 f2 d f1 distance from the last Back focal length (so1=): b.f.l. surface to the second d f1 f2 focal plane f d f Front focal length (s =): f.f.l. 1 2 distance from the first i2 d f f surface to the first 1 2 focal plane Multiple lenses: effective focal length f d f f d f b.f.l. 2 1 f.f.l. 1 2 d f1 f2 d f1 f2 If d 0 (lenses pushed close together), the effective focal length is: 1 1 1 1 ... If d 0 f f1 f2 f3 Fresnel lens Stops and apertures Field stop - an element limiting the size, or angular breadth of the image (for example film edge in camera) Aperture stop - an element that determines the amount of light reaching the image Field stop determines the field of view Apperture determines amount of light only Entrance and exit pupils Entrance pupil: the image of the aperture stop as seen from an axial point on the object through the elements preceding the stop Exit pupil: the image of the aperture stop as seen from an axial point on the image plane through the elements preceding the image no lenses between image and A.S. - exit pupil coincides with it. Chief and marginal rays Marginal ray: the ray that comes from point on object and marginally passes the aperture stop Chief ray: any ray from an object point that passes through the middle of the aperture stop It is effectively the central ray of the bundle emerging from a point on an object that can get through the aperture. Importance: aberations in optical systems Vignetting The cone of rays that reaches image plane frome the top of the object is smaller than that from the middle. There will be less light on the periphery of the image - a process called vignetting Example: entrance pupil of the eye can be as big as 8 mm. Telescopes are designed to have exit pupil of 8 mm for maximum brightness of the image Collection Efficiency Which lens collects more light? f = 10 mm f = 10 mm Relative aperture Entrance pupil area determines the amount of light energy that reaches the image plane. Typically pupil is circular: the area varies as square of its diamater D. The image area varies as square of dimensions and is ~f2. Flux density varies as (D/f)2. (D/f) relative apperture f-number Example: f=50 mm, D = 25 mm: f/# = 2 f f /# denoted as f/2 D F-number The F-number, “f / #”, of a lens is the ratio of its focal length and its diameter. f / # = f / d f f d1 f f d2 f / 1 f / 2 The F/# f f /# D •referred to as the “f-number” or speed •Exposure time is proportional to the square of the f-number •measure of the collection efficiency of a system •smaller f/# implies higher collected flux: • f or D decreases the flux area • f or D increases the flux area f-number of a camera lens Change in neighbouring numbers is 2 Intensity is ~1/(f/#)2: changing diafragm from one label to another changes light intensity on film 2 times Numerical Aperture Another measure of a lens size is the numerical aperture. It’s the product of the medium refractive index and the marginal ray angle. NA = n sin() Why this definition? Because the magnification can be shown to f be the ratio of the NA on the two sides of the lens. High-numerical-aperture lenses are bigger. Numerical Aperture NA nsin •describes light gathering capability for: lenses microscope objectives (where n may not be 1) optical fibers … NA photons gathered Depth of Field Only one plane is imaged (i.e., is in focus) at a time. But we’d like objects near this plane to at least be almost in focus. The range of distances in acceptable focus is called the depth of field. It depends on how much of the lens is used, that is, the aperture. Object Out-of-focus Size of blur in plane out-of-focus Image f plane Focal Aperture plane The smaller the aperture, the more the depth of field. Depth of field example A large depth of field isn’t always desirable. f/32 (very small aperture; large depth of field) f/5 (relatively large aperture; small depth of field) A small depth of field is also desirable for portraits. Phys 322 Mirrors Lecture 13 Ancient bronze mirror Liquid mercury mirror Hubble telescope mirror Planar mirror also called plane, or flat mirrors s = -s i r i o Sign convention: s on the object side is positive, and negative on the opposite side Planar mirror For a plane mirror, a point source and its image are at the same distance from the mirror on opposite sides; both lie on the same normal line. Image is virtual, up-right, and life-size (MT = +1) yi si The equation for lens works: MT yo so.
Load more

Recommended publications
  • Breaking Down the “Cosine Fourth Power Law”
    Breaking Down The “Cosine Fourth Power Law” By Ronian Siew, inopticalsolutions.com Why are the corners of the field of view in the image captured by a camera lens usually darker than the center? For one thing, camera lenses by design often introduce “vignetting” into the image, which is the deliberate clipping of rays at the corners of the field of view in order to cut away excessive lens aberrations. But, it is also known that corner areas in an image can get dark even without vignetting, due in part to the so-called “cosine fourth power law.” 1 According to this “law,” when a lens projects the image of a uniform source onto a screen, in the absence of vignetting, the illumination flux density (i.e., the optical power per unit area) across the screen from the center to the edge varies according to the fourth power of the cosine of the angle between the optic axis and the oblique ray striking the screen. Actually, optical designers know this “law” does not apply generally to all lens conditions.2 – 10 Fundamental principles of optical radiative flux transfer in lens systems allow one to tune the illumination distribution across the image by varying lens design characteristics. In this article, we take a tour into the fascinating physics governing the illumination of images in lens systems. Relative Illumination In Lens Systems In lens design, one characterizes the illumination distribution across the screen where the image resides in terms of a quantity known as the lens’ relative illumination — the ratio of the irradiance (i.e., the power per unit area) at any off-axis position of the image to the irradiance at the center of the image.
    [Show full text]
  • Galileo and the Telescope
    Galileo and the Telescope A Discussion of Galileo Galilei and the Beginning of Modern Observational Astronomy ___________________________ Billy Teets, Ph.D. Acting Director and Outreach Astronomer, Vanderbilt University Dyer Observatory Tuesday, October 20, 2020 Image Credit: Giuseppe Bertini General Outline • Telescopes/Galileo’s Telescopes • Observations of the Moon • Observations of Jupiter • Observations of Other Planets • The Milky Way • Sunspots Brief History of the Telescope – Hans Lippershey • Dutch Spectacle Maker • Invention credited to Hans Lippershey (c. 1608 - refracting telescope) • Late 1608 – Dutch gov’t: “ a device by means of which all things at a very great distance can be seen as if they were nearby” • Is said he observed two children playing with lenses • Patent not awarded Image Source: Wikipedia Galileo and the Telescope • Created his own – 3x magnification. • Similar to what was peddled in Europe. • Learned magnification depended on the ratio of lens focal lengths. • Had to learn to grind his own lenses. Image Source: Britannica.com Image Source: Wikipedia Refracting Telescopes Bend Light Refracting Telescopes Chromatic Aberration Chromatic aberration limits ability to distinguish details Dealing with Chromatic Aberration - Stop Down Aperture Galileo used cardboard rings to limit aperture – Results were dimmer views but less chromatic aberration Galileo and the Telescope • Created his own (3x, 8-9x, 20x, etc.) • Noted by many for its military advantages August 1609 Galileo and the Telescope • First observed the
    [Show full text]
  • Geometrical Irradiance Changes in a Symmetric Optical System
    Geometrical irradiance changes in a symmetric optical system Dmitry Reshidko Jose Sasian Dmitry Reshidko, Jose Sasian, “Geometrical irradiance changes in a symmetric optical system,” Opt. Eng. 56(1), 015104 (2017), doi: 10.1117/1.OE.56.1.015104. Downloaded From: https://www.spiedigitallibrary.org/journals/Optical-Engineering on 11/28/2017 Terms of Use: https://www.spiedigitallibrary.org/terms-of-use Optical Engineering 56(1), 015104 (January 2017) Geometrical irradiance changes in a symmetric optical system Dmitry Reshidko* and Jose Sasian University of Arizona, College of Optical Sciences, 1630 East University Boulevard, Tucson 85721, United States Abstract. The concept of the aberration function is extended to define two functions that describe the light irra- diance distribution at the exit pupil plane and at the image plane of an axially symmetric optical system. Similar to the wavefront aberration function, the irradiance function is expanded as a polynomial, where individual terms represent basic irradiance distribution patterns. Conservation of flux in optical imaging systems is used to derive the specific relation between the irradiance coefficients and wavefront aberration coefficients. It is shown that the coefficients of the irradiance functions can be expressed in terms of wavefront aberration coefficients and first- order system quantities. The theoretical results—these are irradiance coefficient formulas—are in agreement with real ray tracing. © 2017 Society of Photo-Optical Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.56.1.015104] Keywords: aberration theory; wavefront aberrations; irradiance. Paper 161674 received Oct. 26, 2016; accepted for publication Dec. 20, 2016; published online Jan. 23, 2017. 1 Introduction the normalized field H~ and aperture ρ~ vectors.
    [Show full text]
  • Te2, Part Iii
    TERMINOLOGIA EMBRYOLOGICA Second Edition International Embryological Terminology FIPAT The Federative International Programme for Anatomical Terminology A programme of the International Federation of Associations of Anatomists (IFAA) TE2, PART III Contents Caput V: Organogenesis Chapter 5: Organogenesis (continued) Systema respiratorium Respiratory system Systema urinarium Urinary system Systemata genitalia Genital systems Coeloma Coelom Glandulae endocrinae Endocrine glands Systema cardiovasculare Cardiovascular system Systema lymphoideum Lymphoid system Bibliographic Reference Citation: FIPAT. Terminologia Embryologica. 2nd ed. FIPAT.library.dal.ca. Federative International Programme for Anatomical Terminology, February 2017 Published pending approval by the General Assembly at the next Congress of IFAA (2019) Creative Commons License: The publication of Terminologia Embryologica is under a Creative Commons Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0) license The individual terms in this terminology are within the public domain. Statements about terms being part of this international standard terminology should use the above bibliographic reference to cite this terminology. The unaltered PDF files of this terminology may be freely copied and distributed by users. IFAA member societies are authorized to publish translations of this terminology. Authors of other works that might be considered derivative should write to the Chair of FIPAT for permission to publish a derivative work. Caput V: ORGANOGENESIS Chapter 5: ORGANOGENESIS
    [Show full text]
  • Panoramas Shoot with the Camera Positioned Vertically As This Will Give the Photo Merging Software More Wriggle-Room in Merging the Images
    P a n o r a m a s What is a Panorama? A panoramic photo covers a larger field of view than a “normal” photograph. In general if the aspect ratio is 2 to 1 or greater then it’s classified as a panoramic photo. This sample is about 3 times wider than tall, an aspect ratio of 3 to 1. What is a Panorama? A panorama is not limited to horizontal shots only. Vertical images are also an option. How is a Panorama Made? Panoramic photos are created by taking a series of overlapping photos and merging them together using software. Why Not Just Crop a Photo? • Making a panorama by cropping deletes a lot of data from the image. • That’s not a problem if you are just going to view it in a small format or at a low resolution. • However, if you want to print the image in a large format the loss of data will limit the size and quality that can be made. Get a Really Wide Angle Lens? • A wide-angle lens still may not be wide enough to capture the whole scene in a single shot. Sometime you just can’t get back far enough. • Photos taken with a wide-angle lens can exhibit undesirable lens distortion. • Lens cost, an auto focus 14mm f/2.8 lens can set you back $1,800 plus. What Lens to Use? • A standard lens works very well for taking panoramic photos. • You get minimal lens distortion, resulting in more realistic panoramic photos. • Choose a lens or focal length on a zoom lens of between 35mm and 80mm.
    [Show full text]
  • Introduction to CODE V: Optics
    Introduction to CODE V Training: Day 1 “Optics 101” Digital Camera Design Study User Interface and Customization 3280 East Foothill Boulevard Pasadena, California 91107 USA (626) 795-9101 Fax (626) 795-0184 e-mail: [email protected] World Wide Web: http://www.opticalres.com Copyright © 2009 Optical Research Associates Section 1 Optics 101 (on a Budget) Introduction to CODE V Optics 101 • 1-1 Copyright © 2009 Optical Research Associates Goals and “Not Goals” •Goals: – Brief overview of basic imaging concepts – Introduce some lingo of lens designers – Provide resources for quick reference or further study •Not Goals: – Derivation of equations – Explain all there is to know about optical design – Explain how CODE V works Introduction to CODE V Training, “Optics 101,” Slide 1-3 Sign Conventions • Distances: positive to right t >0 t < 0 • Curvatures: positive if center of curvature lies to right of vertex VC C V c = 1/r > 0 c = 1/r < 0 • Angles: positive measured counterclockwise θ > 0 θ < 0 • Heights: positive above the axis Introduction to CODE V Training, “Optics 101,” Slide 1-4 Introduction to CODE V Optics 101 • 1-2 Copyright © 2009 Optical Research Associates Light from Physics 102 • Light travels in straight lines (homogeneous media) • Snell’s Law: n sin θ = n’ sin θ’ • Paraxial approximation: –Small angles:sin θ~ tan θ ~ θ; and cos θ ~ 1 – Optical surfaces represented by tangent plane at vertex • Ignore sag in computing ray height • Thickness is always center thickness – Power of a spherical refracting surface: 1/f = φ = (n’-n)*c
    [Show full text]
  • Compound Light Microscopes Magnification
    Compound Light Microscopes • Frequently used tools of biologists. • Magnify organisms too small to be seen with the unaided eye. • To use: – Sandwich specimen between transparent slide and thin, transparent coverslip. – Shine light through specimen into lenses of microscope. • Lens closest to object is objective lens. • Lens closest to your eye is the ocular lens. • The image viewed through a compound light microscope is formed by the projection of light through a mounted specimen on a slide. Eyepiece/ ocular lens Magnification Nosepiece Arm Objectives/ • Magnification - the process of objective lens enlarging something in appearance, not Stage Clips Light intensity knob actual physical size. Stage Coarse DiaphragmDiaphragm Adjustment Fine Adjustment Light Positioning knobs Source Base Always carry a microscope with one hand holding the arm and one hand under the base. What’s my power? Comparing Powers of Magnification To calculate the power of magnification or total magnification, multiply the power of the ocular lens by the We can see better details with higher power of the objective. the powers of magnification, but we cannot see as much of the image. Which of these images would be viewed at a higher power of magnification? 1 Resolution Limit of resolution • Resolution - the shortest distance • As magnifying power increases, we see between two points more detail. on a specimen that • There is a point where we can see no can still be more detail is the limit of resolution. distinguished as – Beyond the limit of resolution, objects get two points. blurry and detail is lost. – Use electron microscopes to reveal detail beyond the limit of resolution of a compound light microscope! Proper handling technique Field of view 1.
    [Show full text]
  • Sample Manuscript Showing Specifications and Style
    Information capacity: a measure of potential image quality of a digital camera Frédéric Cao 1, Frédéric Guichard, Hervé Hornung DxO Labs, 3 rue Nationale, 92100 Boulogne Billancourt, FRANCE ABSTRACT The aim of the paper is to define an objective measurement for evaluating the performance of a digital camera. The challenge is to mix different flaws involving geometry (as distortion or lateral chromatic aberrations), light (as luminance and color shading), or statistical phenomena (as noise). We introduce the concept of information capacity that accounts for all the main defects than can be observed in digital images, and that can be due either to the optics or to the sensor. The information capacity describes the potential of the camera to produce good images. In particular, digital processing can correct some flaws (like distortion). Our definition of information takes possible correction into account and the fact that processing can neither retrieve lost information nor create some. This paper extends some of our previous work where the information capacity was only defined for RAW sensors. The concept is extended for cameras with optical defects as distortion, lateral and longitudinal chromatic aberration or lens shading. Keywords: digital photography, image quality evaluation, optical aberration, information capacity, camera performance database 1. INTRODUCTION The evaluation of a digital camera is a key factor for customers, whether they are vendors or final customers. It relies on many different factors as the presence or not of some functionalities, ergonomic, price, or image quality. Each separate criterion is itself quite complex to evaluate, and depends on many different factors. The case of image quality is a good illustration of this topic.
    [Show full text]
  • Chapter 3 (Aberrations)
    Chapter 3 Aberrations 3.1 Introduction In Chap. 2 we discussed the image-forming characteristics of optical systems, but we limited our consideration to an infinitesimal thread- like region about the optical axis called the paraxial region. In this chapter we will consider, in general terms, the behavior of lenses with finite apertures and fields of view. It has been pointed out that well- corrected optical systems behave nearly according to the rules of paraxial imagery given in Chap. 2. This is another way of stating that a lens without aberrations forms an image of the size and in the loca- tion given by the equations for the paraxial or first-order region. We shall measure the aberrations by the amount by which rays miss the paraxial image point. It can be seen that aberrations may be determined by calculating the location of the paraxial image of an object point and then tracing a large number of rays (by the exact trigonometrical ray-tracing equa- tions of Chap. 10) to determine the amounts by which the rays depart from the paraxial image point. Stated this baldly, the mathematical determination of the aberrations of a lens which covered any reason- able field at a real aperture would seem a formidable task, involving an almost infinite amount of labor. However, by classifying the various types of image faults and by understanding the behavior of each type, the work of determining the aberrations of a lens system can be sim- plified greatly, since only a few rays need be traced to evaluate each aberration; thus the problem assumes more manageable proportions.
    [Show full text]
  • Section 10 Vignetting Vignetting the Stop Determines Determines the Stop the Size of the Bundle of Rays That Propagates On-Axis an the System for Through Object
    10-1 I and Instrumentation Design Optical OPTI-502 © Copyright 2019 John E. Greivenkamp E. John 2019 © Copyright Section 10 Vignetting 10-2 I and Instrumentation Design Optical OPTI-502 Vignetting Greivenkamp E. John 2019 © Copyright On-Axis The stop determines the size of Ray Aperture the bundle of rays that propagates Bundle through the system for an on-axis object. As the object height increases, z one of the other apertures in the system (such as a lens clear aperture) may limit part or all of Stop the bundle of rays. This is known as vignetting. Vignetted Off-Axis Ray Rays Bundle z Stop Aperture 10-3 I and Instrumentation Design Optical OPTI-502 Ray Bundle – On-Axis Greivenkamp E. John 2018 © Copyright The ray bundle for an on-axis object is a rotationally-symmetric spindle made up of sections of right circular cones. Each cone section is defined by the pupil and the object or image point in that optical space. The individual cone sections match up at the surfaces and elements. Stop Pupil y y y z y = 0 At any z, the cross section of the bundle is circular, and the radius of the bundle is the marginal ray value. The ray bundle is centered on the optical axis. 10-4 I and Instrumentation Design Optical OPTI-502 Ray Bundle – Off Axis Greivenkamp E. John 2019 © Copyright For an off-axis object point, the ray bundle skews, and is comprised of sections of skew circular cones which are still defined by the pupil and object or image point in that optical space.
    [Show full text]
  • Depth of Field PDF Only
    Depth of Field for Digital Images Robin D. Myers Better Light, Inc. In the days before digital images, before the advent of roll film, photography was accomplished with photosensitive emulsions spread on glass plates. After processing and drying the glass negative, it was contact printed onto photosensitive paper to produce the final print. The size of the final print was the same size as the negative. During this period some of the foundational work into the science of photography was performed. One of the concepts developed was the circle of confusion. Contact prints are usually small enough that they are normally viewed at a distance of approximately 250 millimeters (about 10 inches). At this distance the human eye can resolve a detail that occupies an angle of about 1 arc minute. The eye cannot see a difference between a blurred circle and a sharp edged circle that just fills this small angle at this viewing distance. The diameter of this circle is called the circle of confusion. Converting the diameter of this circle into a size measurement, we get about 0.1 millimeters. If we assume a standard print size of 8 by 10 inches (about 200 mm by 250 mm) and divide this by the circle of confusion then an 8x10 print would represent about 2000x2500 smallest discernible points. If these points are equated to their equivalence in digital pixels, then the resolution of a 8x10 print would be about 2000x2500 pixels or about 250 pixels per inch (100 pixels per centimeter). The circle of confusion used for 4x5 film has traditionally been that of a contact print viewed at the standard 250 mm viewing distance.
    [Show full text]
  • Lab 11: the Compound Microscope
    OPTI 202L - Geometrical and Instrumental Optics Lab 9-1 LAB 9: THE COMPOUND MICROSCOPE The microscope is a widely used optical instrument. In its simplest form, it consists of two lenses Fig. 9.1. An objective forms a real inverted image of an object, which is a finite distance in front of the lens. This image in turn becomes the object for the ocular, or eyepiece. The eyepiece forms the final image which is virtual, and magnified. The overall magnification is the product of the individual magnifications of the objective and the eyepiece. Figure 9.1. Images in a compound microscope. To illustrate the concept, use a 38 mm focal length lens (KPX079) as the objective, and a 50 mm focal length lens (KBX052) as the eyepiece. Set them up on the optical rail and adjust them until you see an inverted and magnified image of an illuminated object. Note the intermediate real image by inserting a piece of paper between the lenses. Q1 ● Can you demonstrate the final image by holding a piece of paper behind the eyepiece? Why or why not? The eyepiece functions as a magnifying glass, or simple magnifier. In effect, your eye looks into the eyepiece, and in turn the eyepiece looks into the optical system--be it a compound microscope, a spotting scope, telescope, or binocular. In all cases, the eyepiece doesn't view an actual object, but rather some intermediate image formed by the "front" part of the optical system. With telescopes, this intermediate image may be real or virtual. With the compound microscope, this intermediate image is real, formed by the objective lens.
    [Show full text]