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OPTI-201/202 Geometrical and Instrumental © Copyright 2018 John E. Greivenkamp 1-1 Section 1 Introduction Foundations of Geometrical Optics OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-2 From National Research Council Report: “Harnessing ” Optics is theencompassing the field physical phenomena ofwith associated science the generation, anddetection transmission, and engineering utilization manipulation, of light. Optics OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-3 e to be finite. He and parabolic . He was tion and suggested that the angle of He believed that vision involved that vision believed He objects using convex , and he lenses, and objects using convex y. The lenses were made of natural noted that light travels in straight lines noted on for a that and showed test of all possible paths. Optica crystal. positive objective and a negative eye . studied the magnification of small attributed to the rainbows of sunlight from individual raindrops. aware of . His work was translated into Latin and was of spherical aberration. His work aware scholars. available to later European was proportional to the angle of incidence. spherical atmospheric refraction, and the actual path was shor rays going from the eye to the object. from the eye rays going and described the law of refraction. 15901608 Zachariusmicroscope. (Netherlands) constructed a compound Jensen Lipperhay Hans (Netherlands) constructs the first telescope using a 1268-1289 invention of eyeglasses in Ital The 140 AD140 Ptolemy (Alexandria) studied refrac 1267 Roger Bacon (England) considered th 1000 Alhazenlenses, by (Bazra) investigated magnification produced 100-150 BC Hero (Alexandria) studied reflecti In the beginning… “Let there be light” BC~300 Euclid (Alexandria) in his History of Optics OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-4 . , he presents the use of Opticks Dioptrics ite light is composed of individual ite light is composed first reflecting telescope to solve the opes and microscopes. He proposes microscopes. He opes and s the achromatic doublet to correct reporting the moons of Jupiter in 1610. reporting the moons e principle of least time for determining the design of a telescope using two positive lenses. of a telescope using two the design angle of incidence and the refraction. using . colors by in lenses. achromatic doublet. positive and negative lenses in telesc ray paths. problem of chromatic aberration. wavelets propagating in an all-pervading ether. upon begins astronomical investigations, 1733 Chester More Hall (England) invent 16211657 Willebrordthe (Netherlands) discovers the relationship between Snell 1668Fermat (France) states th Pierre de Issac (England) constructs Newton the 1758 John and Peter Dollond (England) patent and commercialize that 16661678 Issac (England) proves that Newton wh 1704theory light (Netherlands) publishes his wave based Christian Huygens Issac (England) publishes Newton his book, 1611Kepler Johannes in his book, (Germany) 1609 Galileo Galilei (Italy) constructs a version of Liperhay’s telescope and History of Optics OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-5 A Treatise on Electricity and ons relating electric and magnetic ssful in detecting the motion of ve nature of light by demonstrating ttering theory to describe the blue e photoelectric effect on the basis that photoelectric effect on e ents the relationships describing the the of a black body source by body the spectrum of a black which contains his four equati which interference. intensity and polarization of reflected and refracted light. reflected and polarization of intensity and circular aperture. Magnetism fields. earth through the stationary luminiferousearth through ether. body source. color of the sky. color of the sky. requiring a quantum of action (Planck’s Constant). requiring a quantum light is quantized into what are now called photons. are now into what light is quantized 1821Jean Fresnel (France) pres Augustin 18011823 Thomas Young (England) proves the wa 18351873 Joseph Fraunhofer (Germany) publishes his theory of . Airy (England) calculates George the diffraction by a pattern produced publishes James Clerk Maxwell (England) 18791887 Thomas Alva Edison (US) develops the electric light. Morely Michelson and (US) are unsucce 18961899 Wilhelm Weinthe spectrum of a thermal or black (Germany) described Lord Rayleigh (England) develops sca 1900 Max Planck (Germany) explains 1905explain th Albert Einstein (Germany) History of Optics OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-6 the process of stimulated emission. p216/local_copies/history_of_optics/hist.htm give rise to the visible laser. ruby. neon. helium and laser using Portions of this history are taken from R. Victor Jones: Portions of this history are taken R. Victor from http://people.seas.harvard.edu/~jones/a 196019611990Maiman Theodore (US) constructs the first laser using a rod of synthetic Ali Javan, W. Bennett and Donald Herriott (US) demonstrate the first gas The Hubble Space Telescope is launched. 193219481958(US) invents polarizing film Land (Polaroid). Edwin Dennis Gabor (Hungary) invents holography. the principles that will (US) propose Charles Townes and Arthur Schawlow 1916proposes Albert Einstein (Germany) History of Optics OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-7 Mostly theory Mostly systems OPTI-201R: OPTI-202R: ster course covering the basics of image . These courses form the foundation of the courses form the foundation . These the topics to be covered include: the topics to be covered And many more…. Lenses and mirrors Location and magnification of images Stops and pupils Prisms Cameras Telescopes and microscopes Optical materials and diffraction Aberrations and image Lens design using computers Lasers and optical testing Interferometry and image sensors Detectors and Radiometry Fiber optic communications formation and optical system design and layout and optical system design formation and of program. Some Optical Engineering OTTI-201R and OPTI-202R constitute a two-seme Future courses in the Optical Engineering program will cover: in the Optical Engineering Future courses OPTI-201R/202R OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-8 Reflection Refraction Optical design Aberrations Radiometry Geometrical Theory: diffraction, polarization diffraction, and quantum effects. E & M Interference Diffraction Polarization in the limit of short . in the limit of short wavelengths. Wave Theory: - properties (location, magnification) imaging - aberrations (image quality) - radiometry (how much light) Energy levels Probability densities Photons Geometrical optics is the study of light study Geometrical optics is the as rays. Light propagates Geometrical optics usually ignores interference, Geometrical optics includes: Quantum Theory: Theories of Optics OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-9 is the distance is the  The between peaks on the wave. between 8  wave. wave. The electric and magnetic fields are 2.99792458 10 m/s  c the direction of propagation. the direction of propagation. In vacuum, the EM wave propagates a the speed of light: a the speed propagates the EM wave In vacuum, or transverse to Light is a self-propagating electro-magnetic What is Light?? OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-10 At a great distance from the source, will a portion of the spherical wave appear to be a plane wave. Distant Point Source produce an expanding spherical wave. A s not propagate, but the wave does. but the wave not propagate, s Point Source There is a direct analogy to water waves, which are also transverse waves. The are also transverse waves. which to water waves, There is a direct analogy –water moves up and down water doe the is a surface of constant propagation time from the source. constant propagation surface of is a wavefront A point source in a homogeneous medium will point source in a homogeneous A Propagation and OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-11 m 1/sec or Hz of the wave is the of the wave   Vm/sec 

 

 Units: V   V    T 1 T in vacuum: in vacuum:

Vc  

 that pass a point in one second. If the wave is propagating with a velocity or speed V, the frequency V, speed velocity or with a is propagating If the wave number of cycles or wavelengths number The time for one wavelength to pass is The frequency is then Wavelength and Frequency OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-12 e portion of the spectrum, but more generally 8  Hz < ~ Light Visible 750 nm for < ~ Light Visible 0.75 µm for 14    nm m 550 0.55 (Green) 5.45 10 2.99792458 10 m/s   

 c  ~ 350 nm < ~ 0.35 µm < optics includes both the ultraviolet infrared portions of the spectrum. optics includes both and Example: The field The of optics often implies only the visibl Electro-Magnetic Spectrum OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-13 Hz 8 14 nm m nm m 367 0.367 550 0.55 (Green) 1.993 10 m/s 5.45 10  1.5 c n     M 0 

 n   V V change as this is determined by the source this is determined by as change slows down in a medium compared to its in a medium compared slows down c V Example: 8 0

 n 2.99792458 10 m/s Speed of Light in Vacuum of Light Speed Speed of Light in Medium of Light Speed    M c

n  speed in a vacuum. not does that in a medium, the frequency Note oscillation. The wavelength changes with the index. Speed of light in vacuum Index of Refraction n The index of refraction tells much light how Index of Refraction OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-14 m3.4 m4.0   Germanium 13 @ vacuumheliumhydrogenairwaterfused silicaplasticsglassborosilicate crown crown glasslight flint glassdense barium crown 1.0 dense flint glass 1.000036 1.000132 diamond 1.51 Silicon 1.46 1.000293 1.33 1.62 1.48-1.6 1.52 @ 10 1.57 1.72 2.4 Typical Indices of Refraction OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-15 optics and paraxial optics. Small optics and angle the image is superposition of intensity of the optical system. These aberrations of the optical system. These includes the effects of aberrations on the on are sometimes included in the analysis. ndependently radiating point sources. Each ndependently tical systems, or optical systems without system, even when perfectly manufactured. when system, even pplied to radiation of any wavelength. pplied to radiation of any Imaging properties (image location and magnification) Radiometric properties Preliminary system design and layout Image quality Optical design source is infinitesimally is point source small; interference. Each there is no independently imaged through the system, and patterns from all of the point images. First-order optics is the study of perfect op aberrations. Analysis methods include Gaussian approximations are used. Aberrations are the deviations from perfection are inherent to the design of optical Additional aberrations can result from manufacturing errors. Third-order optics (and higher-order optics) effects of diffracti system performance. The a be Geometrical optics principles can Any object is comprised of a collection Any of i Geometrical Optics OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-16 b b d ds n n(s) a a the time it takes light to propagate from the time it takes light to propagate a vacuum in the same time a vacuum the light travels () () () ds b a a b   Vs  a b c 1   T Tnsds OPL nd   OPL n s ds point a to point b along a ray. point a to b along medium: In a homogeneous that light travels in is the distance OPL The distance d in the medium. The Optical Path Length OPL is proportional to Optical Path Length OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-17 z Image nt to the image point is constant. on and are to the wavefront surfaces. are normal to the wavefront and on example is or Gradient Index materials. GRIN a first-order optical system: • wavefronts are spherical All or planar. • OPL along each ray from The the object poi Source If n is not constant, the rays may be bent. An Wavefronts are surfaces of constant OPL from a source point. from a source Wavefronts are surfaces of constant OPL propagati direction of energy indicate the Rays If n is constant, the rays are straight lines. In a perfect optical system or Wavefronts and Rays Wavefronts and OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-18 lines in a material since a of constant index L of adjacent paths. The actual ray path is nt a to point b through any set of media is nt a to point b through any set of media is nt a to point b through any set of media um (a minimum or a maximum) with respect ll of the light rays connecting a source point to r than a curved connecting the points. rst to other paths closely approximation, the one that takes the least time. the one Correct Form: The path taken by a light ray in going from poi in the fi equal, that renders its OPL the one adjacent to the actual path. extrem is either an actual ray of the OPL The of adjacent paths or equal to the OP to the OPL stationary with respect to closely adjacent paths In a first-order or paraxial imaging system, a its image have equal OPLs. straight are Fermat’s principle implies that rays points is shorte two straight line connecting A method for determining valid ray paths. A Popular form: The path taken by a light ray in going from poi Fermat’s Principle OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-19 b n x x C – an imaging lens B B (a and b are object and image) b are object and (a and distance to the edge of the lens. distance to the edge A AC a lens (nt) compensates for the extra for the lens (nt) compensates The extra OPL through the center of the The extra OPL Equal OPL OPL OPL x x b b b C B B AC A –mirror a concave – a – an elliptical reflector M a a a M piece of string fixed length with its fixed at the foci. ends An ellipse with a can be drawn (a and b are the foci of ellipse) (a and Equal OPL Maximum OPL Minimum OPL Fermat Examples OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-20 The wavefront spacing is different and the across the boundary, must meet at the wavefronts interface. The Law of Refraction or Snell’s Law can be easily derived from this condition. The same arguments and analysis of Reflection. lead to the Law Refraction and Wavefronts Refraction and OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-21 p b 2 2 2  n 2 h y L 1 1 L 1 h 1  n p - y This is, of course, Snell’s Law Snell’s This is, of course, a 2  sin 2 1

  sin 2 1 2 0 path for a valid ray  py py     yy 2   2 22 1 hpy hy 2 22 12   dL dL  2  1 12 22 nn 11 dy L 11 2 2 dy L   dL dL Lhy Lhpy  

   sinsin sin 0 sin dy dy dy 1122 1122 dOPL Refraction at a Surface Use Fermat’s Principle to determine the valid Fermat’s Use The ray ray path across a refractive boundary. variable y The from point a to b. goes defines the ray intersection at interface: nn nn Then OPL n L n L OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-22 2  2 is conserved. n  1 n 1  slope and angles at the ray intercept are the incident ray, the refracted ray and the the refracted ray and the incident ray, are the angles 2 

 and 1  are the indices of refraction in  2

 sin sin and n 2211 1 nn Snell’s Law of Refraction Snell’s Law between the rays and the surface normal. the rays and between the two media and the two For refraction at a surface, where n An important second part of this law is that part of important second An surface normal must be coplanar. surface, the surface For refraction at a curved used. When propagating through a series of parallel interfaces, the quantity n sin OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-23 . The . 1 , the = - n n 2 c V  1 2   Speed of Light inSpeed of Light Vacuum Speed of Light in Medium Light of Speed  n a special case of Snell’s law where n law where a special case of Snell’s ity instead of speed in the definition of ity instead of speed a function of propagation direction.

  21  index of refraction is now also negative. of refraction is now index Following a reflection, light propagates from right to left, and its velocity can be can its velocity left, and right to from reflection, light propagates a Following considered to be negative. Using veloc The law of reflection is often considered sign of the index of refraction is now sign of the index For reflection at a surface, the angle of incidence equals the angle of reflection: Both angles are measured relative to the surface to sign the minus sign is due normal, and conventions. again, the two rays and surface normal Once coplanar. must be Law of Reflection OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-24 = 1.0 2 for n Critical angles 1.31.41.5 50.3° 1.6 45.6° 1.7 41.8° 1.8 38.7° 1.9 36.0° 2.0 33.7° 31.8° 30.0° total internal reflection um to a low index medium, the angle of eam is completely reflected. This process eam ce. For a sufficiently ce. large incident angle, y is reflected at the surface by :

 2 1 n n 2 1 n n  C   21

 critical angle sin sin and  1122 12 sin sin 1 nn nn 1  sin sin The incident angle that just satisfies incident angle this The inequality is the Incident rays above the critical angle undergo the critical Incident rays above angle undergo TIR; rays at angles less than the critical angle are refracted. refraction is larger than the angle of inciden of the angle refraction is larger than solution to Snell’s law: there is no There is no solution when Under this condition, the light ra (TIR). There is no refracted beam and the b in prisms high reflectivity. reflection, is often used to achieve achieves 100% and When a ray propagates from a high index medi from a high index a ray propagates When Critical Angle and Total Internal Reflection TIR OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-25 Refracted Reflected 1 1 < n n 2 n is about 42°. Even at angles below the the ray that is reflected. The ratio of the ray that is reflected. The Incident e Fresnel reflection coefficients, and the e Fresnel reflection coefficients, and systems with steep surface slopes; TIR instead systems with steep surface slopes; TIR on, the reflectance of an interface is 2 e critical angle is approached.   21 21 nn nn    

 At normal incidence, with no absorpti Care must be taken in the design of optical Care must be taken in the design critical there is a significant angle portion of th by is governed transmission reflection to reflectance increases rapidly as th at glass/air interfaces. occur of refraction may For common (n = 1.5), the critical glasses (n = 1.5), For common angle Partial Reflection OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-26 Small Gap FTIR Some Reflection tends a short distance into the lower index ndex surface is brought in close proximity to ced. The transmitted The intensity is a strong ced. ce in providing a switching mechanism from ce in providing must be a fraction of a wavelength. TIR Frustrated TIR (FTIR) Frustrated TIR This phenomenon has practical has importan This phenomenon When a ray undergoes TIR, the electric field ex i If another high wave). medium (evanescent the TIR interface, a transmitted ray is produ size. The gap function of the gap high to low transmission two prisms by pushing together. OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-27 x  y Distances: UP: DOWN: Negative RIGHT: Positive Positive LEFT: Negative Angles (measured from the +x axis): COUNTER-CLOCKWISE:CLOCKWISE: Positive origin is the reference location for distances. The x-axis is the reference direction for angles. The Negative In a real-world measurement, the origin can be placed anywhere. The reference direction for angles can be any direction. Cartesian Coordinates OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 3-28 x  y Negative ise, it is common to use what is known as is known what to use it is common ise, e of the paper), it of the paper), is a negative angle. e ane of the paper) it is a positive angle. x nd along the reference direction. nd  fining the signs of angles. y Positive the “right-hand rule” for de right ha Place the fingers of your fingers towards the desired direction. Curl your (or out of the pl thumb points up If your (or into the plan thumb points down If your the right-hand rule is completely consistent with Cartesian measurements and The clockwise and counter-clockwise conventions. Rather than using clockwise and counterclockw Right-Hand Rule OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 3-29 x x  y y  x x   y y All of these angles have positive values. All of these angles have the tail reference direction is defined by of the arrow. The Changing Reference Directions OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-30 tric optical system is the axis rence point, line, or plane in a Cartesian se angles are positive; clockwise angles +z) in a medium with a positive index of +z) in a medium with positive index fined to be the directed distance from its fined rence line or plane in a Cartesian sense right are positive; below or to the left right are positive; are below nt reference definitions and sign conventions rected distances and angles are identified rected distances and by and variables to be easily related the diagram and tion following a reflection are reversed. the reference point, line, or plane. • axis of symmetry The of a rotationally symme and is the z-axis. •relative distances are measured to a refe All or to the sense: directed distances above negative. •relative angles are measured All to a refe (using the right-hand rule): counter-clockwi are negative. • radius of curvature a surface is The de vertex to its center of curvature. • travels from left to right (from -z Light refraction. •signs of all indices refrac The Reference Definitions and Sign Conventions To aid in the use of these conventions, all di aid in the use of these conventions, To arrows with the tail of the arrow at Throughout this course, a set of fully-consiste This allows the signs of results is utilized. or to the physical system. OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-31 z  P  F f F   z Negative Values  z  u  h z z nds on the reference. For example, a ray nds R R z the optical axis or a surface normal. u z h  z  F  Values Positive   R f   P Sign Conventions The value and sign for an angle or position depe sign for an angle value and The angle can be measured relative to either OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-32 Set the equation up for the Set the equation up the When figure as drawn. signs of quantities change, are still the equations valid is the correct answer and obtained. 0 0 0     CAB CAB B    fined from the head of A to the head of B to the head A of fined from the head C is defined from the tail of A to the head of B to the head A from the tail of C is defined If the direction of B changes to negative, If the direction of B changes the same equation holds: CABA   0   CABA C B B B C C C A A A A B C Redefine the reference for A: C is now de is now A: C the reference for Redefine If A changes direction to negative, the same equation holds: the same equation direction to negative, changes A If Use of Sign Conventions OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-33 hz hz hz z h z h z h   ce locations (apex of the angle;  000  000  000     tan   tan 0  tan 0 h ds independent of the signs quantities. ds independent   are defined from the same referen z z  z  for this figure?  h h What about these variations? What about What is tan Use of Sign Conventions for Angles In all of these Figures, z, h and base of the triangle), and the same equation hol OPTI-201/202 Geometrical and Instrumental Optics © Copyright 2018 John E. Greivenkamp 1-34 hz hz zz hh z h    000  000    tan   tan required equation. It is essential to know equation. required nce changed to the opposite side of triangle: nce changed h h z z   If the quantity z is redefined to have its refere If the quantity z is redefined to have The choice of the reference location changes the choice of the reference location changes The the reference location for all quantities. Choice of Reference