Geometrical optics Design of imaging systems

Geometrical optics Design of imaging systems Geometrical optics is either very simple, or else it is very complicated Richard P. Feynman What s it good for? 1. Where is the image?. How large is it? 3. How bright is it? 4. What is the image quality? 1 Useful when size of aperture is > 100 Why? Robert McLeod 18 1 P. Mouroulis & J. Macdonald, Geometrical Optics and Optical Design, Oxford, 1997

Ray and eikonal equations E The ray equation Approx. solution of ME when n(r) is slow jk0s r r Er e S k k r S r n k r x k y k Assume slowly varying amplitude E and phase S Substitute into isotropic wave equation and retain lowest terms z 0 x y z E.g. plane wave r k x y 0S 0 z E.g. spherical wave Ray equation reduction of Maxwell s equations. Countours of S(r) at multiples of Ray = curve to S(r) n(r) S nr S(r) Optical path length [m] Robert McLeod 19 B A

Ray and eikonal equations The eikonal equation An equation for evolution of ray trajectory Take square-root of ray equation, dr dr Sr nr sˆ nr then take derivative in s, s ˆ d d dr S r nr r r dr d and chain rule for, S r S r s Parametric distance along ray [m] r s Ray trajectory [m] dr finally apply another identity. 1 n r S r d S n r r S n r r S r Robert McLeod Saleh & Teich 1.3, Born & Wolf 7 th edition, pg. 19 0 dr 1 n r n r Ray n r nr

Ray and eikonal equations Ray trajectory n=constant Finally! d d r n n s r 0 dr c s dr n n 0 r 0 Eikonal n(r) = n o, a constant where c is a constant. Thus, we have discovered that, in homogenous materials, rays travel in straight lines! Robert McLeod 1

Rays and refraction History of geometrical optics 80BC Euclid's Catoptrics light goes in straight line. However light was emitted by eye (not) law of reflection was correctly formulated in Euclid's book Hero of Alexandria, in his Catoptrics (first century BC), also maintained that light travels with infinite speed. adopted the rule that light rays always travel between two points by the shortest path 1000 AD Arab philosopher Alhazan (Abu'ali al-hasan ibn alhaytham) first person to realize that light actually travels from the object seen to the eye 1676 Olaf Römer demonstrated that light must have a finite velocity, using his timings of the successive eclipses of the satellites of Jupiter 100-170AD Claudius Ptolemy rough law of refraction was studied experimentally (only close to normal incidence) 161 Dutch mathematician Willebrord Snell comes up with correct law using sines. Does not publish 1637 French philosopher René Descartes was the first to publish, French countries call Snell s law Descartes law of refraction 1658 French mathematician Pierre de Fermat demonstrated that all three of the laws of geometric optics can be accounted for on the assumption that light always travels between two points on the path which takes the least time Robert McLeod Kevin Curtis OESD lecture notes

Rays and refraction Pinhole camera (1/) Mo Ti, China, invented 5 th century BC Aristotle observes image of eclipse cast through leaves, 300 BC Aristotle formulates theory of light and color (some right, some wrong) Alhazen (Ibn Al-Haytham) invented CA 1000 AD Della Porta invented CA 1600 Kepler named it Camera Obscura and suggested lens for efficiency, 1604 l l Magnification, M = - l / l Very large depth of focus Very small chromatic effects Very poor power efficiency Dr. Webster Cash at CU funded by NASA Institute for Advanced Concepts to examine pinhole camera in space as extrasolar planet imager. Abelardo Morell pinhole image of Manhattan Robert McLeod 3

Rays and refraction Pinhole camera (/) Impact of pinhole size A pinhole camera has no lens but uses a very small hole some distance from the film/screen to produce an image. If we assume that light travels in straight lines,thentheimageofa distant point source will be a blur whose diameter is the same as the pinhole. However, diffraction will spread the beam into the Airy disk. Geometrical blur Diffraction blur Image is sharpest when the geometrical blur (the pinhole size) is equal to the diffraction blur (Airy disk diameter). Design example: What pinhole diameter is optimum for a visible-light camera that is 100 mm deep? D 1. 1. sin D.44 L.44 0.37 mm.5510 3 D / / L 100 D/ L=100 mm D/ Robert McLeod Kevin Curtis OESD lecture notes 4

Rays and refraction Evolution of the eye Lens evolved to improve radiometric efficiency of an otherwise pretty good imaging system. Robert McLeod Wikipedia 5

Fermat s principle Fermat s principle Another (important) form of the eikonal Optical path length nr Propagation time = S / c Hero of Alexandria (in Catoptrica, ca 50 AD): Light travels in straight lines Pierre de Fermat (ca 1650): Light travels the path which takes the minimum time. Correct: The time of travel is stationary: B A n Example 1: Concave mirror with radius of curvature < ellipse Example : Lifeguard problem See Born and Wolf for a derivation from Maxwell s equations Why do we care? At an image point, all OPL must be equal. S r 0 B A P. Mouroulis & J. Macdonald, Geometrical Optics and Optical Design, pg 11-1, Oxford, 1997 Robert McLeod R. Feynman, Lectures on Physics 6

Fermat s principle The lifeguard problem Problem: The lifeguard at point A nee to get to the drowning swimmer at B in minimum time. What is the optimum point at which to enter the water given that the lifeguard runs at speed V/n on sand and swims at speed V/n in water? Sand n A Water n B Solution 1: Write equation for travel time with entry point as parameter, minimize time, solve for entry point. Solution : Use Fermat s principle which says that angles obey Snell s Law, use this constraint to solve for path. Example: If lifeguard on shore, this is a problem of total internal reflection, so is the critical angle: -1 sin n n Sand n 90 A Water n B Robert McLeod 7

Radiometry Radiometry Review of terminoligy & inverse square law I E A R A I R L A Solid angle is area of sphere subtended A over radius of sphere R Intensity I of point source is power emitted into solid angle Irradiance E on surface is power per unit area A Irradiance of surface by a point source is given by intensity I of point source over distance to surface R Radiance of surface normal area A illuminated by a point source of power radiating into a solid angle Q Energy [J] Power (flux) [J/s=W] I Intensity [W/sr] E Irradiance [W/m ] L Radiance (photometric brightness ) [W/(sr m )] Robert McLeod O Shea chapter 3, Mouroulis and Macdonald 5.3 8

Radiometry Examples What is irradiance m from 100 W light bulb? 100 W E R I 100 W m m m W 5 m What is brightness of a typical 100 W light bulb? filament area m 4 Sr W L W/m /Sr] for a bulb with a 1 mm filament Lots of power, moderate area, but huge radiation angle = low brightness. What is brightness of a typical 1 W laser? W L w m M divergence angle rad 0 W 1 w m M rad M 0 w 0 W/m /Sr] for a perfect 1 W laser at = 1um Less power and similar area to the lamp filament, but MUCH smaller radiation angle = high brightness. Besides the potentially narrow spectrum, brightness is what sets lasers apart from lamps. Robert McLeod 9

Radiometry Radiometry via rays Consider a finite source radiating into a cone A 1 A R 1 E A 4R R Irradiance E on surface A given power into cone Now launch a set of rays inside this cone and examine the ray density as a function of radius N N A 4R E A general proof that ray density is proportional to irradiance can be found in Born and Wolf. Robert McLeod Mouroulis and Macdonald 1.4 30

Foundations Postulates of geometrical optics Rays are normal to equiphase surfaces (wavefronts) The optical path length between any two wavefronts is equal The optical path length is stationary wrt the variables that specify it Rays satisfy Snell s laws of refraction and reflection The irradiance at any point is proportional to the ray density at that point Robert McLeod 31