Optoelectric System Design
Optoelectric System Design ECEN 5616
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Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design ParaXia thin lenses and graphical ray tracing What s a lens Paraxial approximation Let us assume for a moment that a lens is a thin phase function that connects all of the rays from an object to an image r t n f 0b 39ect ima e 1 n n y g lens m n t 2 n1 it2 r2 n 1 it r2 Sm r Fermat s Primin 2 2 I r n r n zm n t I 518130 B1norn1alparaX1al t 2 t approx1rnation 2 i 2 r n n r S r Of a paraXial lens lens I 2 t t i 5 2 1 Gaussian thin lens equation f t t f Focal length of lens n1 ilf Power of lens diopters Note that Mouroulis uses K for power Robert McLeod 49 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design ParaXial thin lenses and graphical ray tracing Power of a lens Physical meaning il Inair t The power of a lens is the algebraic increment in curvature added to the incident wavefront Note that this means that two thin lenses in contact are equivalent to a single lens with the sum of the powers Robert McLeod Pr Mouroulis amp Jr Macdonald 352 50 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Paraxial thin lenses and graphical ray tracing Graphical ray tracing Solving Maxwell s Eq with a ruler I 1 Object MELlt0 f l Image 1 A ray through the center of the lens is undeViated 2 An incident ray parallel to the optic axis goes through the back focal point 3 An incident ray through the front focal point emerges parallel to the optic axis and occasionally useful 4 Two rays that are parallel in front of the lens intersect at the back focal plane 5 Corollary two rays that intersect at the front focal plane emerge parallel n Object J Image Robert McLeod 51 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design ParaXial thin lenses and graphical ray tracing Graphical tracing Negative lenses v quot 1 9 y t I I y t Virtual image M E gt 0 y t l A ray through the center of the lens is undeViated 2 An incident ray parallel to the optic axis appears to emerge from the front focal point 3 An incident ray directed towards the back focal point emerges parallel to the optic axis and occasionally useful 4 Two rays that are parallel in front of the lens intersect at the back focal plane 5 Corollary two rays that intersect at the front focal plane emerge parallel Robert McLeod 52 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design ParaXial thin lenses and graphical ray tracing Real and virtual Images and objects project rays back to Real object intersection Virtual image 4 t Virtual object Real Image W If you can t see the light by placing a screen at the plane where it is in focus then it is Virtual Equivalently an image is Virtual if you need another lens e g an eye to make the image real Equivalently real Virtual objects are to the left right of the surface and real Virtual images are to the right left Robert McLeod 53 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design ParaXia thin lenses and graphical ray tracing Tracing mirrors Mirror system Equivalent lens system A I t t lt t lt t t gt v t t i 33quot x quot i t t quot Robert McLeod 54 Design of ideal imaging systems with geometrical optics ParaXia thin lenses and graphical ray tracing Thin lens equations derived forms and quantities ECE 5616 OE System Design Z A Z 4 t gt t lt f lt f gt y M E L 2 t Via similar triangles y t transverse magnification dt dt 0 t 2 2 Take derivative of thin lens equation I 2 dt t 2 z d t t z M Longitudinal magnification f f Z y H Via s1m11artr1angles I I f z y y f M 2 Newton s lens equation ZZ f applies to thick lenses as well Robert McLeod 55 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design ParaXial thin lenses and graphical ray tracing Angular magnification 4 t gt t u N y M E l 2 t From previous slide y t Angular magnification is ratio of two axial rays rays that cross the axis at the object and image 1 h t t l MQE z z Iz hisrayheightatlens u h t t M Remember the radiometric unit L radiance or photometric brightness in units of Wsr m2 We just found that as size of an object goes up it s angular extent decreases by the same amount Brightness is conserved Robert McLeod 56 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design ParaXial thin lenses and graphical ray tracing Scheimpflug condition Imaging from a tilted plane Ray parallel to AB must intersect at back focal plane l s 4 I a l I r 7K B Paraxial firstorder optical systems image lines to lines and planes to planes even when the objects are not normal to the optical axis Proof Draw ray ABA B using a ray parallel to AB through the front focal plane Then draw rays AA and BB Robert McLeod 57 Design of ideal imaging systems with geometrical optics ECE 5616 OE System Design ParaXia thin lenses and graphical ray tracing Throw Assume you need to deliver an image a fixed distance Tfrom the object k lt t gt t y i lt f lt f gt y I T E Z I Eq 1 Throw is sum of object and image dists 1 1 1 Eq 2 Gaussian thin 1ens equation in air I l f t2 IT 0 Eliminate t 1i i 2 i t Ti T 4T and solve fort 2 f Two solutions t and t interchange roles with the minimum throw being T 4fand tt 2f Robert McLeod Mouroulis and Macdonald 3 56 58 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design ParaXial thin lenses and graphical ray tracing Single lens design problem Summary Possible unknowns t t T M f Equations I t 1 1 1 I E l l t t t f So given any two variables we can solve for the remainder The Newtonian form of the thin lens equation can be handy if f and M are given z E t f 2 z a f f fM Commonly in camera sorts of applications T and M are given T a rr f1 f1 M f T1 sH M 2 TM M 12 Work through the examples 32 and 33 to be sure you are comfortable with the sign conventions Do graphical sketches to confirm the equations Robert McLeod 59 Turning ideal imaging systems into real optics ECE 5616 013 system Design maging in nonideal systems Generalized imaging systems Infinite paraxial system glt Entrance pupil Exit pupil Consider an ideal infinite aperture imaging system The electric field at the output plane must be related to the input field by I I I 1 x Eix yiniM Our goal in this section is to understand how the image field for coherent light and intensity for incoherent light differs from this perfect copy in the presence of diffraction and aberrations Robert McLeod Goodman Fourier Optics first edition Chapter 6 205 Turning ideal imaging systems into real optics ECE 5616 013 system Design maging in nonideal systems Generalized imaging systems Finite paraxial system We can define a function h that describes the response in the output plane X y for an infinitely small excitation in the input plane Xy Since we know that Maxwell s equations are linear the response to a general input must be the sum of the responses to every point in the input plane E x y I Mx y x yEx ydxdy Now let us assume that the imaging system is shift invariant For example there is no vignetting The superposition integral can now be written E x y I Jhx Mx y M yEx ydxdy Note that this is in the form of a convolution integral This reduces to the infinite paraxial case if whim We know from linear systems theory that a convolution in real space can be represented by a product in Fourier space so it must be possible to describe the shiftinvariant imaging system via E kwky HkxkyEkxky where H is the Fourier transform of h and kX 2n fX is spatial fr n Robert nggd Cy 206 Turning ideal imaging systems into real optics ECE 5616 013 system Design maging in nonideal systems Frequency response of a diffractionlimited system This result can be formally derived from Fourier optics but we can argue our way there from what we know about paraXial optics 27 ky 7 11 6 27139 y Entrance pupil Pemxy 1 inside y 0 outside pupil y tfy Rays that pass pupil thus satisfy PXyl or HifxfyPm Arfxirfykem MM13 Robert McLeod 207 Turning ideal imaging systems into real optics ECE 5616 013 system Design maging in nonideal systems Coherent diffraction Iimited imaging Coherent transfer function Impulse response H 03013 gitytmigi hx y E1Hfxfy Hifx fyi emi rfmfy 2 J1 27r rD21t 2 mi 27 rD2tr Let s find the first null of the impulse response 2715338317 or re 06i 2it NA The cuto frequency highest spatial frequency that passes is f 3 NA 0 2M xi Robert McLeod 208 Turning ideal imaging systems into real optics ECE 5616 013 system Design maging in nonideal systems Incoherent light To understand the concept of coherence define the mutual coherence function which is the crosscorrelation of the electric field at different spatial locations GemsEnrEerrgt where lt gt indicates ensemble average but can be considered an infinitetime average in most cases When E is wellcorrelated with itself across the entire object G is a constant the light is referred to as coherent At the other extreme when even a small change of position Irlr2gtt causes the correlation to sharply decrease G approaches zero the light is completely incoherent There are many interesting and important intermediate cases but we will limit ourselves to just the two extremes The intensity is related to G by 17 0070 2 ltE7tE7tgt Examples of coherent light 1 Lasers 2 Lamp after spatial filtering This leads to the common but not totally accurate definition of spatial coherence as light that appears to emerge from a single point 3 Expanded laser beam passed through a static diffuser which is a violation of the previous rule Incoherent light virtually anything else lamps stars blackbody laser passed through randomly timevarying spatial diffuser Robert McLeod Saleh and Teich Chapter 10 209 Turning ideal imaging systems into real optics maging in nonideal systems Imaging with incoherent light Substitute our impulseresponse equation into the definition of intensity and interchange the orders of integration I3939 7 zE 7 zgt ECE 5616 OE System Design I Idildylj Idizdyzhbc 3519 y y1hx 3529 y y2ltE3El7 y1E3527 where 36 E Mx and we are assuming quasimonochromatic light For incoherent light the mutual coherence function is nearly zero outside a region roughly one wavelength in diameter We can approximate this as a delta function ltE561y1E562y2gt z K1061 i1 gt602 y E which reduces the integral to I gt 00 I 2 1r K W xy y Ir W61 which says that incoherent systems are also linear in intensity and that the impulse response is the abs magnitude squared of the coherent impulse response Using a FT rule again we can write the Optical transfer function I kxky Hkx kyIkx ky Hkk Fhxy2Hkk Hkk Robert McLeod 210 Turning ideal imaging systems into real optics ECE 5616 QB system Design maging in nonideal systems Properties of the OTF if Coherent transfer function H fxfy Autocorrelation operation Hkxky Hkxky X Area of overlap between displaced copies Optical transfer function Himy IOTFI Modulation transfer mction Robert McLeod 21 1 rTummg ideal imaging systems into veal optics 55515 Ogsysm new rlmagmg in nonrideal systems Generalized pupil function Imaging with aberrations Ptevious analysis used pupil function that Was eithet 0 light blocked o1 1 light passed this Was the only nonrideal element in the otheIWise petfect pataxial optical imaging system Now considet adding abenations to the pupil as wavefront 277m Pallx y Pallx ylexpli jkwlx yll k L 1 that abenations neVeI inctease the MTF modulus of the OTFA OTF vsl Waves of defocus Goodman mm 5 Wm 41m a Hit Mr N a willing mm m n mm mm Romecmd 212 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Radiometry Radiometry of point sources Inverse square law Q A Solid angle S2 is area of sphere R2 subtended A over radius of sphere R2 I Intensity I of pointw is power 1 Q emitted into solid angle 2 E Irradiance E on surface is A power 1 per unit area A L z Radiance of surface of normal area A by a QA oint source radiatin into solid an le S2 2 0 Radiance of a tilted surface Qcos 9A A0089 Q Energy J 1 Power ux JszW I Intensity Wsr E Irradiance Wrnz L Radiance photometric brighntess Wsr m2 Nate MampM distinguish between E irradiance an a receiver and M radiant emittance afa source will use Efar bath Robert McLeod O Shea chapter 3 Mouroulis amp Macdonald 5 3 1 78 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Radiometry Extended sources Lambert s law Irradiance of plane by point source as function of angle I Q E 6 Acos 6 Acos 6 RCOSG ACOSG I A R 2 I cos 2 200836 A A cos 6 R R E 0cos3 6 i Radiance of extended source L AQ Lw 15 L0 Radiance vs angle for A 2 cos 6 cos 6 isotropic source 6 g 6 Lambert s Law 9 cos Intensity falls off like cos away from normal L6 M L0 Radiance vs angle for A 149 COS 9 Lambertian source Robert McLeod 179 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Radiometry Power emitted by a Lambertian source and captured by a lens Calculate incremental power dCIgt radiated from tilted area A into cone of solid angle d9 dcp L6A cos add L6A cos Sid L r L6A cos 6 r2 sin 6d6d 2 r L6A d6d Now integrate over all i and finite aperture angle 60 to or 2 sin 26 lt1gt j jL0A d6d 0 0 2 L75 A sin2 0 Lambertian source L6L L75 hNA Let area A be a circle of EszHZ radius h field height n 1 Power collected by a lens is given only by radiance of source and H2 POWER COLLECTED IS GIVEN BY H2 2 H is conserved Power can not be increased Thus radiance can not be increased CONSTANT RADIANCE THEOREM Robert McLeod Mouroulis amp Macdonald 533 and 534 1 8O Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Radiometry Imaging extended sources Constant brightness theorem again Entrance pupil Exit pupil l Conservation of energy LIAQ Definition of brightness LIM Linear and angular mag 2 L Constant brightness theorem T Transmission of system 0 lt T lt 1 Robert McLeod 181 Design of ideal imaging systems with geometrical optics ECE 5616 013 System Design Radiometry Blackbody sources Ideal incoherent sources Radiate energy but unless T gt 700 0K emit very little Visible radiation and thus appear black Planck s explanation of the blackbody spectrum in 1900 was the beginning of the development of quantum mechanics The radiance of a blackbody L does not depend on angle and they are thus ideal Lambertian sources 2hc3 1 3 L 7m Wm Sr Planck s equation 9 AM 2 b 28977685 um K Wein s displacement law 2 4 l 2 8 W StephanBoltzman law E OT m2 0 5 6704X10 mzK 4 for total emittance 7 X 1013 j 6 X 1 013 l3 3 5 X 1013 T7000 K Max414 urn E 4 X 1013 T5500 0K Max527 pm 3X10 T wom mf um a 2 x 1013 q 1 x 1013 0 02 04 06 08 l A um So the surface of the sun is roughly 5500 OK and humans radiate in the infrared at about 95 pm Robert McLeod 182 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Radiometry Lasers vs lamps Spectrum Laser Incandescent lamp Intensity counts 4000 E 20 3500 if 3000 339 m 2500 II 2000 E 60 wk 1500 E h 1000 Ian 500 350 550 a o 35 950 95 0 400 500 600 700 800 900 1000 h g39av algnqih nmzl Wavelength nnnn meters Efficiency Luminous e iciency is power in Visible spectrum electrical power in As low as 2 for incandescent bulbs Filters can reduce spectral width but dramatically reduce efficiency Wallplug e iciency is optical power out to electrical power in As high as 60 for diode lasers 30 for diodepumped solid states Brightness MW 2 MW 7 W0 m2M2 mfw radD2 filament area lmZJX475Sr D n 2 105 WmZSr for a bulb 1M2 with a 1 mm2 filament emitting 1012 WmZSr for a perfect 1W 1 W laser at 7 1 um in air Robert McLeod 1 83 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Radiometry Example Radiometry of projector T rm 391 m m 0 H quotU to CD a a o g gtlt a B 3 gtlt g B q 5 o 391 3 0 E 391 0 U Q 8 E D D o quotG g 0 5 O 0 Cr 5 39D m quotU 39D H 0 U 0 3 gt3 5 3 T H H g o 22gt 0 53 in quot3 B D in 0 2w 3 Ha o o v H O 5 O p h i b g l o o D U o O D a v P D 8 g 8 Jquot 8 5 a r V1 0 g 8 5 w 39D 5 VJ If HC lt Hp edges of image will appear dark projection lens is stopped down If HC gt Hp light is lost on entering the projector Typically design for HC just a bit larger than Hp Calculate power on screen from area and spec irradiance E A Calculate H of condensor from power lamp brightnesscp 2lesz Given slide radius this gives NA of condensor H hNA 4 Can now design projection imaging system with this H 93 Robert McLeod 184 Deswgn 01 rPhaseSDace Design Phase space At the object 1000 5 Hm Pix 150mm ihmm 200 H nkukyk k kYk izjioo 10 25 Area 10 05W 2000 500 o x mm 39Design 01 ideal imaging quot 39 39 Design 7P 359 Phase space Before the lens 1000 50 mm D10 mm 5 pm pix D25mm 9L5 pm 7 lt 75 mm 150 mm H nkukyk kz 7kyk xfxzzff 425 200 Q 10 x 25 g Area 5mm 7m 1005um 2000 729 710 x cm 10 2 126 ECE 5616 OE System Design Design oi ideal imaging systems with geometrical optics Phase space Phase space After the lens 1000 5 mm pix 4 75 mm 150 mm 4 Afte Lens H nkukyk nk kyk 1 1 F1 i205 i 75 x f M x 1 20 20 2 0125i375 25 H I Area 2000 7200 7amp00 10 20 Design 01 ideal imaging systems with geometrical optics ECE 5515 013 System Design Phase space At the Fourier plane 1000 5 pm pix 150mm lt 75mm H quotkukyk k kyk x f39xz1f f 1015 2m 20 20 25 3 Area 10mm 7200 2005mm 2000 10 20 128 o x mm 39Design 01 ideal imaging sys se 1000 5 gm pix lt 75mm 39L I Design Phase space At the image 50 mm D225mm A Image I D10 mm 105 pm 150 mm inm 7200 7400 x f xz1ff Hznkukyknk kyk 1 1 7 5 70 20 20 25 Area 10mm 2005um 2000 o x mm Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Delano diagrams Delano diagrams A paraxial design tool AKA yV diagrams yu tracing calculates ray heights through system given lens prescription The Delano diagram calculates the lens prescription given the ray heights The optical invariant H is a scale factor for the entire calculation and thus all lens spacings and powers and the locations of pupils principal planes and focal planes can be adjusted to give a desired resolvable number of spots RObeIT MCLeOd Richard Ditteon Modern Geometrical Optics Chapter 8 1 30 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Delano diagrams Delano diagrams A graphical tool for system design Also known as y y diagrams since we will plot the height of the marginal ray y and the height of the chief ray y as the two rays evolve through the system A 0 mm Dimg 5 mm Dobj 225 mm D 2 AS m O y X TLZ lt 75mm To visualize the approach consider putting the chief ray in the XZ plane while leaving the marginal ray in the y z plane A r 150mm gt 4 o 0 Y X A L z W Note that by convention the object height is negative Robert McLeod 131 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Delano diagrams Delano diagrams Introduction single lens system Now consider a skew ray whose X coordinate is given by the chief ray and whose y coordinate is given by the marginal ray That is the chief and marginal rays are the projections of this skew ray in the xz and yz planes W O l b ts Now we look at this skew ray from the image plane that is we project the ray onto the xy plane Ty 3 Radius of AS Obj height Image height The ray spirals clockwise and can never do otherwise Robert McLeod 132 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Deano diagrams Afocal system Keplerian telescope Paraxial ray diagram l Delano diagram v quotlt Robert McLeod 133 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Deano diagrams Delano diagrams Lenses on the diagram Lenses bend the skew ray In general the ray continuously spirals clockwise Negative lens Virtual image Positive lens Virtual image Y H Postive lens real image w quotlt Not allowed w quotlt Robert McLeod r 134 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Deano diagrams Delano diagrams Conjugate line For ease of drawing lets momentarily draws the chief ray back in the yz plane so we have a 2D drawing Note that we have kept the negative object height convention Now let us shift the object to a new it which is conjugate to a 13 petition But don t draw a new marginal or chief ray Note that the old marginal or chief rays cross both the new object and new image At the new conjugate planes there is a new magnification and the heights of the old marginal and chief rays at the new object unprimed and image primed planes should be linearly related 2 l The new image location 3 y can thus be found Via a Geometrical version of this equation is a line conjugate line drawn through the origin through the origin The new magnification is I 2 old ray heights at new image location 2 old ray heights at new object location Robert McLeod 135 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Deano diagrams How to find distances Since we projected out the z dimension it seems unlikely that the Delano diagram can tell us about z locations of elements It turns out that it can with a little manipulation Let s start by finding the reduced distance between elements in terms of y and ybar yk1 yk 142d Marginal ray transfer eq yk 1 2 yk 17261 Chief ray transfer eq Combine the two equations yk1yk12 yk1yk kdk yklyk ukdk and solve for the reduced distance dz kakn yklyk yklyk kakn k ukyk1ukyk1 H Multiply this by nk to get actual distance So if we know H y and ybar give the element separation How would we specify H Remember that at an object or image plane y0 37 limits the field and u is the numerical aperture Then L xi L xi H 2 may 2 NA 6 z spam 2 r0 2 2 Note that we took y negative so that H lt 0 in this convention Robert McLeod Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Delano diagrams Calculation of distances from the yybar diagram Radius of ASlZ5 3 V Obj height25 Image height5 Define the skew ray position vector in the xy plane at a surface k 77 ykje y k 7 Any two such points and the origin form a triangle The area of that triangle can be found by 1 d d l A Akk1 ark X rk1 Eyk1yk YkYk1Z Comparing to the equation for separation distance we find 2nkAkk1 H 1 21e125 25 31 25 In our example d 2 39 75 0 1 12575 25 0416667 d So d between k and kl is given by Ak k1 kk1 Note that both A and H are negative s1nce r0 r100 proceed clockw1se 321 320 125 This gives the initial ray angle no 2 016667 0101 75 Robert McLeod 137 Design of ideal imaging systems with geometrical optics ECE 5616 OE System Design Deano diagrams How to find lens powers We now know the ray heights and ray angles Using a similar approach to how we found d we can find the lens power I Mk 2 uk yk k Marginal ray refraction eq Mk 2 LTk yk k Chief ray refraction eq Combine and solve for i I ukuk ukuk 1 H Thus we have found the distance between lenses the ray angles and the lens power just from y and y at each surface This is the essence of the Delano method On the following pages we ll use a system roughly like this this is just a sketch and not guaranteed to be exactly the system on the remaining pages but has roughly the right properties f1 Robert McLeod 138 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Deano diagrams Finding the pupils Use the conjugate line I Entrance pupil 2 image of AS in object space As on conjugate line page extend skew ray until in intersects the conjugate line The intersection gives y and ybar of the entrance pupil Note that ybar is zero as it should be for the pupil Aperture Stop R1624 Field ht in obj space 5 3 Field ht in img space So REmPupil 10 IfH 02 then 1 1 2quot0 yEm Pupil yr yr yEm Pupil 21 10 2 60 d 100 lEm Pupll H y 02 A Exit pupil 2 image of 8 in image space SO RExitPupil 639 1 1 2quot0 Y2 yExitPupil yExitPupil 3 2 21 40 61 30 H 02 Robert McLeod 139 d 2ExitPupil Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Delano diagrams Finding the principal planes Extend the object and imagespace rays until they cross Note that this now looks like a single refraction our definition of PPs We can draw a conjugate line through the intersection thus we have found the conjugates of unit magnification How do we find the locations of P and P 2n0ypi1 y1vpl 218 2 6 1l d 2 H 2 02 2 50 2ltn0 ypaz yzypi 2181 4 1l 39 2 H 2 02 2 60 5 Robert McLeod 140 Design of ideal imaging systems with geometrical optics Deano diagrams Finding front and back focal length ECE 5616 OE System Design Draw the conjugate line parallel to the imagespace ray The image is now at infinity thus the objectspace point is the front focal plane 2n0 m1 ml 21 5 2 6 25 H 2 02 2 25 A conjugate line between the object space and image space with the image space intersection at infinity Draw the conjugate line parallel to the objectspace ray The object is now at infinity thus the imagespace point is the back focal plane 2a ypa my 21 31 405 H 2 02 15 A conjugate hne between the image space and Object space with the object space intersection at imcinity Robert McLeod 141 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Deano diagrams Finding the effective focal length The EFL is the distance from the front focal point F to the front principal plane P or the distance from the back principal plane P to the back focal point F Note that F PP F O is a parallelogram so the area of the two triangles is guaranteed to be equal therefore the front and back EFL are identical as required A 2ltnogtiiypfF mp 218 25 5 1l 75 H O2 F dFP Starting from H and the height of the marginal and chief ray at each surface we have found the complete prescription for the system including all lens sizes locations and power the locations of pupils principal planes and the effective focal length Robert McLeod 142 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Behavior of lenses and curved mirrors Real surfaces Refractive via Fermat s principle Curved refractive surface aka lens n n 1 z z Paraxial power of curved E n n C n n refractive surface Robert McLeod Mouroulis and Macdonald 33 67 Design of ideal imaging systems with geometrical optics Single and compound lens systems Real surfaces Refractive via Snell s law ParaXial Snell s Law Replace refraction with ray angles Paraxial approximations to angle obeying sign conV Cancel y rearrange Robert McLeod ECE 5616 OE System Design 68 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Behavior of lenses and curved mirrors Lensmakers equation Not that anyone would make a lens with it In a few weeks we ll trace a twolens system and derive a general expression for how to combine the power of two lenses separated by a distance d We ll go ahead and use this expression before we ve derived it to get what is commonly referred to as the lensmaker s equation which is a bit of a misnomer since it would be a bad way to make lenses 1 1 2 d 1 2 Power of two optics separated by d L L 112 n 1R1 n Rn 1 1 2 quot11 c1n 1c21 n1c1c2n 12 71 c1 3 c1 CZ 1 for dlt ltR1R2 Lensmaker s equation Why this is a bad lens design equation usually Robert McLeod Mouroulis and Macdonald 35 69 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Behavior of lenses and curved mirrors Real surfaces Reflective Derive power of mirror by placing object at center of curvature Since ray strikes surface normally it must return to the same point Two points to remember 1 t is positive to the left after re ection 2 All quantities as always labeled are 1 2n 2 n C Paraxial power of spherical mirror f Remember that by our sign convention Rlt0 in the geometry shown yielding a lens power which is positive Robert McLeod 70 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Behavior of lenses and curved mirrors Lenses in contact Eg Magnin mirror 61 cZ CI zCDl 612 613 cln l 2nc2 c1 ln uation 31 lt 0 as drawn Derived on previous page 2 lt 0 as drawn Lens equation but using negative indices per sign convention when light travels R gt L a symmetric concave lens with 201n 1 zncz curvatures 31 a convex mirror of curvature 2 Robert McLeod Mouroulis and Macdonald 3 55 71 Optical instruments ECE 5616 OE System Design Prisms Prisms What are they good for Fold Erect or rotate images Change direction of propagation Fold system for compactness Retroreflect Disperse vs 9 Control beam parameters Anamorphic telescopes Vary angle position path length Amplitude or pol division Beam splitters Robert McLeod 185 39Other important optical components 13035616013 System Design Prisms Folding prisms Bouncing pencils to analyze image orientation t f i o 4 V 7 74 Right angle prism Retroreflector Penta prism Compare orientation to plane mirror Even if re ections constant deviation Such prisms are often used with telescopes to erect the image ip upright objective ll I ep V l l i i cquot i v ROMAN object virtual image Same function as RAP but add roof to get extra re ection Image Dove prism inverts image Rotation of prism rotates image at twice the angle Dispersion at first interface correct at output I 3 I Robert McLeod 186 Other important optical components ECE 5616 OE System Design Prisms Tunnel diagrams Tool to simplify raytracing P i Porro erecting prisms very common 2A in binoculars A l T 3D tunnel diagram A l A Ra t t h 1 1 b y race roug g asss a s 2A 2A Replace glass with air and effective thickness paraxial onlyl Insert into paraxial unfolded ray trace 2An 2An Robert McLeod 187 ECE 5616 QB System Design Anamorphic prisms Often better than cylindrical telescope M cos6 cos6 l t Mzh Problems Compression largest near HR 7 tolerances and polarization dependence Angular bandwidth quite low works best for collimated beams Advantages Lower aberrations than cylinders Cheap Robert Mcleod 188 ECE 5616 OE System Design oF assive components Prisms Thin prism tricks Risley prisms Beam is deviated by angle n loc If prism is rotated about its aXis the beam is de ected in a circle Two cascaded prisms n10c give arbitrary Xy deviation For small oc control of deviation can be quite fine Sliding wedge A As above if CC is small control of displacement can be quite fine Focus adjust Variable path length R g W181 and Mounting of Prisms and Small Mirrors SPIE 1 998 1 89 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Ray and eikonaI equations GRIN lenses Common in fibertelecom applications Full pitch 2 imaging lens Half pitch 2 collimating lens Fractional pitch lt5 as typically used Robert McLeod 190 Other important optical components ECE 5616 OE System Design Prisms GRIN lenses via eikonal Good example and important lens technolgy np n01ABjm Power law radial index distribution a Vn0 Plug into eikonal S ilnltpgti7lilnltplldip2ll p 1 dz dz dz Z araX1a approx d d 0 z Simplify with known dependencies 1 mApmivamApmi1 no quot75 Plug in nltPgt d2 Egakz dpz a a P Spe01al case m2 0z 2 p0 cosKz 06 sinK z Solution for ray trajectory Robert McLeod 191 Dem rRay and eikonai equaiions Numerical solution of eikonal For arbitrary index distributions 3D Gradient index distribution XY slice of 5n 1 X2 slice of En Raytrace in X2 Robert McLeod Diffractive optics ECE 5616 OE System Design ntroduction Diffractive optics Introductionterminology 0 Classes 0 Diffractive optical element Modification of the optical wavefront via subdivision and individual modification of the phase andor amplitude of the segments Grating linear segments uniform diffraction angle Computer generated hologram A DOE in which the structure has been calculated numerically Holographic optical element DOE in which the structure is generated by the interference of optical wavefronts Discretization Binary optic phase or amplitude structure with two levels Typically created via a single etch step 0 Dammann grating Binary optics with repetitive pattern generates N beams fan out Multilevel optic Same as binary but with M etch steps to achieve N22M levels Kinoform Phase DOE with smoothly varying profile limit of N gtoo Blazed Grating with linear sawtooth segments Fabrication 0 Direct machining aka ruling or diamond turning fab via mechanical machining Often used for masters Lithography Direct write scan laser or ebeam over photoresist Interference holography inc near field Masks greyscale multiple exposure Replication Robert McLeod O Shea Diffractive Optics Design Fabrication and Test 1 93 Diffractive optics ECE 5616 OE System Design Diffraction gratings Diffraction gratings Basics Real space Fourier space out x E k Am6kx kxim Sinckx eijk0nsinKGt X L FT gt mk0n kX mKG L 2 mk0n e jmKG 2 EM xrect Eour x Equot Xrect i Am 2 1m kon sinckx km mKG g What are angles of diffracted waves 275 km 2 7s1n 6m Conservatron of transverse k k mKG xiin 275 275 s1nt9in m l G s1n 49m s1n 49m m G Gratlng equat10n Robert McLeod 194 Diffractive optics ECE 5616 OE System Design Diffraction gratings Resolving power aka number of spots Real space Fourier space Bred null gblue peak Rayleigh resolvability criterion K 27 L K sm i m G S 1 m G kred blue kred m27rAG 27rL kblue mZEAG algebra 6 A 1 i 1 G 10 mL 20 L R E mN where N A number of grating lines 51 G illuminated Robert McLeod 195 Diffractive optics ECE 5616 OE System Design Diffraction gratings Estimation of grating R Why gratings are interesting Holographic gratings of 1800 lpmm are typical in the visible A 10 mm beam and firstorder diffraction would yield R 2 101800 2 18000 or a minimum resolvable wavelength shift of 03 nm in the visible For a prism at the minimum deviation condition symmetrical incident and exit angles the resolving power can be shown to be VIE AR xtB 170nm 5 dnb R B bv 1nY 1 b In the visible a b 25 mm prism would give resolving power 1100 Crown 3400 Flint or 5 05 to 017 nm roughly an order of magnitude lower resolution than a grating Robert McLeod 196 Diffractive optics ECE 5616 OE System Design Diffraction gratings Bandwidth aka Free spectral range When will diffractions be confused with the neighboring order 9 red m BMW m1 mK m1 K s1n1 Gs1n1 G red kblue 1146 M k red blue 1m lue AA m blue Thus firstorder grating spectrometer could operate from 400 to 800 nm Robert McLeod 197 Diffractive optics ECE 5616 OE System Design Diffraction gratings Efficiency Overview by type 1 Sinusmdal phase km TEZ Thin phase grating It 06 Typically many orders D chm 2K0 7 27 A Q 2 can t reach 100 in any 04 kxiirt 1KG e g smusmdal ii i ii ii 4 Exception blazed phase grating n Can be 100 in single order A An l E gtII Thin amplitude grating Lossy by definition typically many orders Exception Sinusoidal amplitude grating DC and orders Thick phase grating Bragg selectivity can give single order and theoretically 100 DE BUT Robyle Lsegcrllsitive to incident wave unlike thin 198 Diffractive optics ECE 5616 OE System Design Diffraction gratings Multilevel DOES Why you pay for them Phase not physical profiles N22 N24 Noo First 8rder diffraction efficiency vs number of levels D05 D1 Dlj 771sincz7N D2 7 D25 2 8 10 12 14 16 Robert M 199 Diffractive optics ECE 5616 OE System Design DOEs as lenses Diffractive lens design Multilevel onaxis Fresnel r2 f2 Fermat refractive all rays have OPL Diffractive all rays have OPL modulo Ink r Thus refractive is limit of diffractive w m0 f V Spherical converging wavefront What is the radial location of the p h zone for a mth order DOE fabricated with N layser r f2 f pm ON2 OPL of each zone differs 2 2 bym ON rp 2fpmlONpmlON zzfpmgON For2fgtgtp10 What is the radius size of the p h zone 2 rp 2 f pm 0 N 2 Radius of pm and 91th zone rp1 2fp 1m20N r 1 r 2 f mi0 N Take difference rp1 rp er1rp Expand 2 Arlt2rpgt Ar f m o rN 4 ZmZO F N Local grating period Roberyhptgofor minimum feature size N reduces F linearly ouch 200 Diffractive optics ECE 5616 OE System Design DOEs as lenses Diffractive lenses 7 dependence of angles Reading a diffractive optic at N and order m that was designed for A and order m Local grating period AG DOE Si m 27rl A m 27A ft 51116 MM m X Change in angle is perfectly analogous to Sin 6 E n Sin 6 refracting into a slab of index neff Note 6 7 A that this index can be lt 1 mx 2 ne m X n m l e mi f hsin 6 Definition of focal length f hSm 6 l Diffracts to 00 set of focii f f sin 6 fn f ml 2 For neff7 1 each suffers Sin 639 e my spherical aberration 201 Robert McLeod Diffractive optics ECE 5616 OE System Design DOEs as lenses Diffractive lenses 7 dependence of efficiency 12 For a kinoform N 00 hX wl t X A X S X Xml constant OPL by design is ml at A nhx t hx 2 n 1hx Calculate from profile n 1 Substitute hx xi t mg Solve for step height x xi S x m n 1 OPL at shifted xi A n 1 27 27 i n 1 X A A mn1 1 X Phase ati 2 nLl T x rect ejAme jx Z 5x IA Transmission mask l Grazing Twltkxgttsmcllkxml lquotllltltalkxwe Which gives us the diffracted electric field vs angle for a uniform Einc 202 Robert McLeod DMracuve optics ECE 5616 OE System Desigi rDOEs as lenses Diffractive lenses 7t dependence of efficiency 22 Efficiency of a blazed grating designed for wavelength 7 and order m with index n read at Wavelength N and orderm with index n 2 mm llquot ll A n39il If n71 I has 100 theoretical DE in the design order and conveniently 0 in all other orders a 1 a blazed grating 2 1 0 1 2 m 1397 1f n 71 12 ablazed grating 1 A Ate has 405 theoretical DE in the 1 1 design order and an equal amount in the next lowest order An in nite m 1 of orders are present Robert McLeod 203 Diffractive optics ECE 5616 OE System Design DOEs as lenses Hybrid refractiveDOES mt f fW From page 182 Z 152 3M2 If used at same order mzm Z i 153 R B y R Y Find change in power over L E Y VD0E From page 170 VDOE Solve for V B R 5896 4861 6563 2 346 This is a the same for all DOES b negative and c very strong Let s design an achromatic f254 mm BK7 singlet 25 25 6213 06 0 BK7 D0E 1 254 Achromatic conditions D0E 21496695 mm BK7 126769 mm Note the refractive power is nearly unchanged and the DOE is quite weak Robert McLeod 204 Background ECE 5616 OE System Design Maxell s equations Maxwell s equations in differential form BE V X E Faraday s law at gt gt VXH J Ampereslaw at V E 0 Gauss laws V D 0 E Electric field Vm H Magnetic field Arn D Electric ux density Cmz B Magnetic ux density Wbmz J Electric current density Amz p Electric charge density Cm3 Vx Curl lm V Divergence lrn Robert McLeod Background Maxe s equations Constitutive relations Interaction with matter t E 80 J at T Ki6 Dispersive amp anisotropic oo Z f a 50 31 Anisotropic i 808 EU Isotropic Nanmagnetic H E Z 0 U T HTdf lu0 00 E Ohm s Law on j 80 PermittiVity of free space 8854 103912 Fm 8 Dielectric constant no Permeability of free space 4 7t 10397 Hm it Relative permeability o Conductivity Qm Robert McLeod ECE 5616 OE System Design Background ECE 5616 OE System Design Maxe s equations Boundary conditions Fields at sharp change of material These are derived from Maxwell s equations In the absence of surface charge or current I 11 Conservation of transverse electric and magnetic fields e Mm 712 Conservation of normal electric and magnetic ux densities Etl Ht1 Dnl Bnl If A Medium 1 Medium 2 r2 Unit vector normal to boundary I Unit vector transverse or tangential to boundary Robert McLeod Background Maxell s equations Monochromatic fields Expand all variables in temporal eigenfunction basis Fourier Transform Note factor of 2x which can be placed in different locations 1 W ft E fwe dw fw i ft e ia dz Monochromatic fields E transform like time domain fields E for linear operators Et ReEej er Vx z jw VXFIjwl3j VB 0 Vsz Removes all timederivates Monochromatic Maxwell s equations Robert McLeod ECE 5616 OE System Design Background ECE 5616 OE System Design Maxe s equations Monochromatic constitutive relations The reason for using the monochromatic assumption Convolution MW 580J 2 r 1737d7 gt 5502al3 E uO Zia Tl H7d7 gt Ez OZMDITI 00 co 21030610161 Inverse Fourier Transform Note that 8 is now fw amp not ft If 8 is not constant in 0 w J e1 tdt it causes dispersion of pulses 0 00 00 Conditions for lossless materials derived from Poynting vector next 2 2 M 8 is the Herrnitian conjugate Robert McLeod Background Waves in space Wave equation Eliminate all fields but E VxVx z jwaE Take curl of Faraday s law 2 OIUOV X H Magnetic constitutive OZIUOD Ampere s law 2 0 80110 8 39 E Electric constitutive Monochromatic WE VXVxE kg E0 V2Ek2E 0 Scalar simplification k0 Wave number of free space c Speed of light in vacuum wc zitAG lm WE Ins Robert McLeod ECE 5616 OE System Design Fourier propagation ECE 5616 OE System Design Derivation Planewave solution Cartesian eigensolution in homogeneous space a a 39 a 4 E ReE0 e t Assume a planewave solution 0 Note that this is an eigenfunction E 2 E0 6 1 F Similar to monochromatic assumption V a J k removes all space derivatives This transforms the wave equation into three coupled linear equations one for each vector component of E in three variables the three components of k To have a nontrivial solution the determinant of this matrix equation must be zero 13x32 k3 0 Characteristic equation By using Gauss Law we can reduce this from 6th order in n to 4th in n actually second order in n2 A N 312114 12T adj TradeI n2 IZI 0 n Index of refraction kk0 adj A Adjoint of matrix IAI A1 Tr A Trace of matrix 2 Ail Robert McLeod H c Chen Theory of EM Waves McGrawHill NY 1983 18 Fourier propagation ECE 5616 OE System Design Derivation Isotropic and uniaxial 811 g 81 1 Isotropic material 811 n4 2811 n2 8121 0 Characteristic equation n i Ride 11 Four solutions 2 2 811 g 811 Uniaxial material in principal coords 833 IE 2 Cos 6 Sin p c Sin 6 Sin p y Cos p 2 Propagation direction 2 8 8 n4 811 n2 0 Characteristic equation 8115111 0 833C0S I Four solutions COS2 Sin 2 n i 1811i 2 ordinary 811 8 2 extraordinary L33 Robert McLeod 19 Index ellipsoid ECE 5616 OE System Deslgi Founer propaganon rDeHVauon Uniaxial surface amp polarizations Positive uniaxial negtnO 9 x f 3 u o Robert McLeod ECE 5616 OE System Deslgi Founer propaganon rDerwauon Biaxial surface amp polarizations Robert McLeod Fourier propagation ECE 5616 OE System Design Derivation Spatial frequency Basis of Fourier optics VA 1 X fx 5 i Sln 61quot w Spatial frequency in 1m A 20 zon 27 2 2 27r kx 27K fx s1n 49W n s1n 49mm Wave number in 1m Multiply I by h and now the k vectors represent quantum mechanical momentum The incident particle strikes a boundary normal to z which changes its momentum in z but not in Xy Therefore transverse momentum is conserved Robert McLeod 22 Fourier propagation Derivation Cartesian eigensolution at Mspace ECE 5616 OE System Design Foundation of Fourier optics 5m N tran S trans Igzkxfckyysz A X k E 2 E6 6 x y kinc I x kinc Sln 01710 ktrans I x ktrans Sln errans Break transmitted wave vector into normal and transverse components Excitation on boundary Transverse wave vector conserved Problem Given orientation of boundary 2 material 2 and boundary excitation kx how does plane wave propagate into material kz Robert McLeod 23 Fourier propagation ECE 5616 OE System Design Derivation Booker quartic Characteristic equation for Fourier optics gt gt A Assume k kxfcky 2k12 kxy kzZ and solve for kz in terms of kxy yielding new characteristic equation Note that this is now a general 4th order equation k 4 3 2 k 614 Z a3 Z a2 Z a1 Z a0O k k k k 0 0 0 0 The coefficients are given in terms of the known variables AT A a Z 39 8 39 Z a A A 4 kxy kxx ky y gr T W A a3 K 88 z gt N a2z ade Tradj72 ia4 4T T A k2 0 adJ8adJ8 za3 Zadjmdjzrii HQ w 5 Q C II OWE Normalization of k by k0 keeps all quantities near unity and independent of 0 in the absence of dispersion Physically kxyk0 is the numerical aperture Robert McLeod H C Chen Theary afEM Waves McGrawHill NY 1983 24 Fourier propagation ECE 5616 OE System Deslgi rExammes Isotropic refraction aka Snell s law gives ray directions at boundary Emm Eoei w m Real space 1mm nW Sin 9W 72 1 Sin 9 mm mm n Sin 6 Robert McLeod quot Fourier propagation ECE 5616 OE System Deslgi rExammes Total internal reflection aka evanescent waves izkx iwzm Ermm E0e Real space Real part Imag part nm Sin 9 Robert McLeod 2e Fourier propagation ECE 5616 OE System Deslgi rExammes Extraordinary refraction and other fun with crystals 5 Real space 1 so km 50 2 3 Z 7 Sale x 7m 6 3 Robert McLeod Physwca name 0V hgm ECE 5616 OE System Deslgi erne waves Saleh amp Teich 62 O39Shea 422 Fresnel Coefficients Amplitude and phase of waves at boundary E n1 Cos 0 12 Cos 0 sm0 a i E 111 Cos 0 112 Cos 0 sm0 1 Et 2111 C0519 ZSin 1939Cost9 t 5 l l E 111 Cos 19 2 C0519 Sln9 arallel TM r 2COS 1COS Tpjamp Cairw H E n2 Cos 19 111 Cos 19 WHj w E 2111 C0519 ZSin Q39Cos t t E 112 Cos 19 1 Cos 0 sm0 1939Cos19 Robert McLeod 28 Physlcal nature 0t llght ECE 5616 OE System Destgi rPlane waves Fresnel Coefficients Special values 20 4O 60 80 Normal incidence Brewster s angle Physical interpretation of BB Dipoles excited in n2 can not radiate in direction of re ected Wave when it is L Robert McLeod 29 Phymca name 0V hgm ECE 5616 OE System Deslgi erne waves Fresnel Coefficients Phase and TIR 25 n1gtn2 o o m L Q H L L m 5 w Amplitude coeffi cients gt A Phase g GoesHanchen phaseshift 2 0 40 60 80 5 D A 7 for 7 2 gt quot1 Phase of TE electric field on re ection 0 for H1 lt r11 TM has the opposite same Sign lt gt BB Robert McLeod 30 100 00 1000120 1000120 05010210 Fourier propagation ECE 5616 OE System Design Fourier optics in 1 equation Tattoo this on your eyelids Scalar version Etxyz F 1thyEtxy0 e jkzw12xyz txy 39Eiiiifa i is give by the 39The transverse wave vector is I kx ky y 39The Fourier transform is jwr kxxkyy Fm E Jdtjdx dy e 1 1 3 Amt eaten Fm E dejdkxdkye Robert McLeod Saleh amp Teich Chapter 4 32 Physical nature of light ECE 5616 OE System Design Propagation and diffraction First Imaging limitation Bandlimiting propagation in free space Spatial frequencies beyond TIR do not propagate Thus only a few wavelengths after the object the radiated field will be band limited For example a 1D rect function Frect 2 sinc ka 2 sinc7 fx L 2 sincL 39 X l 25 O 8 L5 pm 2 A1nm 15 06 a La 0 4 La 05 FT 02 0 0 705 775 75 725 0 25 5 75 10 775 75 725 0 25 5 75 10 X m kx lMml Ekx 2k00 Gibb s ringing 25 l 2 k0275lLm 08 lt1 1 5 06 E E 1 m 04 IFT m 05 02 0 O 05 775 75 725 0 25 5 75 10 775 75 725 0 25 5 75 10 X um kx lHm The highest spatial frequency that can be transmitted is thus fX 1 orkX 2 W Robert McLeod 33 Fourier propagation ECE 5616 OE System Design Applications Nearfar boundary Rate of diffraction due to curvature kXI Akz Fourier space Real space How far will the light from a rectangular aperture propagate before it begins to diffract Calculate the phase accumulated between DC and first null and assume when this reaches 7 then the beam will look significantly different Phase at 13 null Phase difference accumulated Phase at C 5 k 2i zz kz kx 0Z kz kx L Z vsr zbetween cenmr and 151 null koz z Isotropic kZ 1 2 H 2 kOZ Z k0 7 Binomial expansion 2k0 20 Liz z 71 At null spatial frequencies now 139 out of phase Now solve for z Where this is true 39 Z 7 Rayleigh range aka nearrfieldfarrfield boundary Thus a bandlimited image can be transmitted directly through Rob free space for a distance lt L2 7r 39Physical nature of light ECE 5616 GE System Design Propagation and diffraction Diffraction from a 1D lens Optical Field 2500 A 1250 E 3 J 00 A 72 3 125i0 00 15 29 43 58 Distance mm a 2368 FRectxLL SinckXL2 g kx 2 k0 Sin9 go 2294 First nullis at 8 Xf NL5800 1200229 a 1721 5 0 1147 E O 1 0574 f VM 7 m A i V i Vr i 7 1250 00 1250 2500 Width Mm Robert McLeod 35 Phi310339 nature 0f light ECE 5616 DE System Design Propagation and diffraction Airy isk The diffraction limited resolution Q What is the electric field at the focus of a uniform amplitude cone Real space Fourier space Transform I gt I39 gt Er2lr2rSin HOJrzfsm 60 Inverse Fourier Transform lt I x i x x AAvrz sln o lfr 15 10 V 1t 15 x1 27 g Sln 60 EiW2i i 11r0 Peak E Energy in ring 383171 10 839 701559 0017 71 101735 00041 28 133237 00016 15 1 2 71 Sin 60 3 39 8 3171 Diameter of first null D 2r 13922 3111 60 Robert McLeod 36 39 Fourier Propagation ECE 5616 OE System Design Relationship to other methods Relationship to Fresnel Convolution with impulse response The transfer function of free space is a 2 2 2 E0kxkyz e JZ E kx ky Eiakxky0i The paraXial approximation to this is found by expanding the square root in a binomial series Note that this is the solution to the paraxial wave equation Helmholtz equation Hakxkyz 2 2 x y jZ 2k jkoz Hakxkyzz e o The transverse spatial inverse Fourier transform of this is the paraxial monochromatic impulse response of free space X2y2 22 jk0 e jko hw x y Z Fk 1k Hw kx ky Z x y 27rz Yielding the Fresnel diffraction formula jkox f2yI2 Emm e jk zllEw w0e zz clde 27L39Z Robert McLeod 37 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design The eye An ato my era serrata m remaquot temporal Clllar39f body iris choroid cornea aqueous humour optic nerve limbus sclera r a ESE Horizontal Section Through Right Eye Roughly a sphere of 12 mm radius Typical extreme range of vision is 380 nm to 740 nm The rods are sensitive to weak light inoperative in strong light and have maximum sensitivity at about 507 nm Rods cover the retina The cones are sensitive to strong light insensitive to weak light and have a maximum sensitivity at 555 nm Cones occupy only the fovea Cones and rods on retina are waveguides Cats back these with a reflective tapetum to get double pass but eyes become cat s eye retrore ectors Pupil diameter changes from 4 to 8 mm many times less that 106 dynamic range of eye Reason is not light reduction but aberration reduction by stopping down the system At any one time dynamic range of eye is 103 Spacing of rods on fovea is about equal to diffraction limited spot size of the pupil at the minimum diameter Most refraction occurs at the cornea large index contrast while the lens adjusts via change of shape to change total power Typical visual resolution is about 6 minutes of arc 2020 vision ability to resolve 5 arc minute features at 20 feet RObert MCLeOd httpwwwduedujcalvertopticscolourhtmEyes 60 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design The eye 2020 A measure of Visual Acuity 20 XX implies that a subject can identify a letter at 20 what a standard observer can at XX feet 20 10 GOOD 20 40 BAD The retina can support better than 20 10 I 1 are min At 2020 Robert McLeod 61 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design The eye The eye A simple optical model for focus I Hgt I I I 555 I I I I 39 1667 w I 2222 739 I I I I I I I 39 l275 I I I I I I i l I 39 I I I l I I I Asphericity P Y 12 p73 n0 abic 39 n 1333 D x 2139 if tmmPrI UIIIC hawiengih of the model r 539 n 1 refractive index for Pmmelrullir u39m elrnglh III 333 Z Z v a 39 I Ll39rnL39lIVD Intlrx rm nmelmwgthII1I nlcmns 320535 Dloptefs Illi all i 0004685 11 1 1 r VH 02M 102 z t 00 f of ocular medium I II 71191 ll iii 5025 39lnrax39 lrirliu of curvature r 5 55 min t Z n f mm VH shape parameter 00 eiliplical ercemricily r Irpl 00325 axial pnsltlnn oi the pllvsical pupil i39mm the apex z 33 mm axial pnslllnn quhe EIIIITIIICL pupil lmm llle apex 2 i5 II39iIII focal power for emmetropir wavelength f u nul W 60 dimmers anterior i39i39x al length for einnwlrupic WIIVGIE IIQIII 667 mm quot03 posierml focal length for cmmelrnpic In39zivelc ngil rlfi39 22 22 mm IIIIIIFIXI RobeIT McLeod Source httpresearchoptindianaeduLibraryCh47ModeEyeCh47ModelEyepdf 52 Desigi Jhe eye Most important quantity Retinal magnification factor aka focal length sinnuni9inradians Robert McLeod 83 Design of ideal imaging systems with geometrical optics ECE 5616 OE System Design The eye Accommodation Accumula un Min lm I EMllmg l hlrm lyuljeds fmdjshntuljecb Object Distance m Focal Length mm Power diopters1m 025 159 628 1 167 599 3 169 592 100 170 588 Infinity 170 588 Power of accommodation 4 diopters in young decreases with age Near point an is 25 cm in young and increases with age as power of accommodation decreases Robert McLeod httpWWWphysicsclassrooIncomClassrefrnU14L6chtml 64 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design The eye Ray tracing the eye Simple instrument single lens magnifier fe TQQ h Near point an25 cm h Gaussian lens amp mag eqs enp 1 715 i h 5 39gtgt Define visual h K magnification MV as 6 MV he increase in image size caused by new lens 4 te gt 39tM fr i ii Mvhe te tM te f6 fM netM Subtract Divide i i 1 DH t D f M v Visual magnification M np M tM of single lens at near 1 up point of eye f M O Shea 1 5 65 Robert McLeod Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Single and compound lens systems The magnifier again via angles useful for infinite conjugates f8 h 0c f 4 f h f For a equal focal lengths fe Visual M V E magnification should be a proportional to ratio of angles an Via similar triangles tM D 1 P Via lens power equation f M quotI7 fM For an object at infinity this becomes M V E Robert McLeod 66 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Paraxia raytracing Paraxial ray tracing Derivation of refraction amp transfer equations Pk Mk M A C I tk tk k l yk Paraxial tangents uk uk k Substitute into thinlens equation 7 k y k u Mk y k k Refraction equation yk uk fkl d I I yk1 y k ukdk lt dk1 Transfer equation yk Height of ray at surface k MampM use hk m pk Power of kth surface diopters uk Paraxial ray angle incident on surface k radians u Paraxial ray angle exiting surface k radians d Total distance between surface k and kl m Robert McLeod 72 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Paraxia raytracing Paraxial ray tracing One method to treat different indices Dealing with different indices of refraction n sin 6 n sin 6 Snell s Law nu z n u Paraxial approximation 1 E nu Reduced angle variable n n 1 Gauss1an thin lens equation I l f Reduced distance variables We can now write equations involving angle and distance but ignoring changes in index Whenever we deal with problems with several difference indices we simply make the above substitutions Robert McLeod 73 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Paraxia raytracing yu tracing A tabular method fobj8 mm feyepiece2 mm feye K C A C T dg W Tube length 160 v v 0 l 2 3 Question to be answered what is the front working distance of a 20X f8mm objective when used with a 160 mm tube as shown 1 Fill in what you know 2 Fill in what you don t know using paraXial refraction and transfer equations Surface k 0 1 2 3 System k 1f 0 18 0 12 d 84 168 2 H 0 849 0 1a Axial ray yk u 10f 12d 12b o a Chosen arbitrarily b Refraction 0 u 1 6 Transfer 0 yl 168 C Transfer 1 y2 92 f Refraction no 84 d Refraction 2 ul 00 g Transfer 84 0 10d6 Robert McLeod O Shea 23 74 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Single and compound lens systems The telescope Keplerian Shown in the afocal geometry df1f2 Relaxed eye focuses at 1m thus telescope are usually not afocal Analysis simpler however Afocal system has no I 1 power ray II to OA 4 0 i c h does not intersect OA hl quot 39 2 in image space hi f 1 f1 f2 a V B h M E Definition of angular magnification 6 0 h 2 L Via similar triangles 1 f 2 l W This is both important and fundamental Robert McLeod 75 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Single and compound lens systems The telescope Galilean f1 quotf2 More compact upright X x quot image Same afocal h1 v A hz condition df1f2 h h f 1 f1 quotf2 V X a B lt d 4 M 9 E g 0 f Note that formula is 2 1 h identical to Keplerian f1 f 2 This is the advantage y of the sign convention M Robert McLeod 76 Design of ideal imaging systems with geometrical optics Single and compound lens systems Microscope f f eyepiece AT obj 4 D tube length Standard tube length is 160 mm Focal system Form image at infinity for simplicity of analysis Visual magnification of instrument is product of linear magnification of objective and Visual magnification of eyepiece M MMM v microscope v eyepiece Note eqs are approximate tube an 1tube gtgt fobj an gtgt feyepice fobj feyepiece Mobj fobj mm Typical NA 4 30 010 10 16 025 20 8 040 60 3 085 100 18 13 Robert McLeod ECE 5616 OE System Design 77 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Single and compound lens systems Overhead projector Mirror ips parity so speaker and Viewers see same image Projection lens must be at field work over a range of image distances and Screen is white diffuse re ector to send light into large achromatic Design angle can be simplified by illumination system Platen Illumination system gives uniform directed white illumination Robert McLeod 78 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Paraxia raytracing ABCD matrices Matrix formulation of paraxial raytracing Mi 1 Oyk E Rkyk Refraction equation Mk k 1 uk Mk I Z k yk k 1 Yk1 dk E Tk Transfer equation uk1 O 1 uk uk yk1 3 1 uldl in bk A d Ml l K K T 39YK1 d0 gt MK1 v J V N y y y 1EM 1 Systan MK ul ul matrix yK1 TlRlIO 3 E N y Conjugate MKH MO uo matrix Robert McLeod 79 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Paraxial raytracing Properties of M N A B 2 AD BC 1 Determ1nantl C D lRiTiMiN i1 iMi Write out the matrix equation for N 37191 N 11370 N 12 K1 N 21370 N 22 If planes 0 and Kl are conjugates final ray height does not depend on initial ray angle N 12 O Conjugate condition If plane 0 is the object space focal plane the slope at the exit plane depends only on the object height N 22 0 Object at front focal plane If plane Kl is the image space focal plane the imagespace ray height depends only on the entrance angle N 11 0 Image at rear focal plane If the system is afocal the direction of the imagespace ray depends only on the direction of the objectspace ray N 21 O Afocal condition Robert McLeod 80 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Paraxial raytracing Use of matrices M N Find image plane given object f h 11 pk I I l K 1 Yo 0 Y1 T 39YK1 d1 K1 N TK1MT0 391 d A B 1 d1 1 1 C D 0 1 AdC BdD d1AdIC C D aC A dgC O Conjugate condition C D aC A B d N 12 0 gives the image location ac D Eg single lens d11 0 1 1 gt d1 1 dK d1 Robert McLeod 81 Design of ideal imaging systems with geometrical optics Paraxia raytracing Form of N And EFL first thick lens concept MEENHAd C IfN120then 3 0 N 11is the magnification 1 N 22 Determinant l M F 1 yo a Effective focal length amp system power MK1 u 0 Tu1 v 0 Eg single lens uK12N21y0N2 N 39391Rl39390 v 0 N I 1 t tt 21 1 r M O z39 N 2 z 0 ltIgt 1M 4 RobeitMcLeod ECE 5616 OE System Design 82 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Paraxial raytracing Optical invariant At objectimage plane special case n u f I n u h Tquot gtlt y c 39 c 1 t t J h h h quot l n l ParaXial Snell s Law y Iquot l Triangles u l h nl nu M y Substitute into M h n l n M n M In a cascaded system Mhkn0 0n1 1 nk kn0 0 h 0 quotDuo quotll 1 nkuk nkuk H nkukhk no lquotoho A conserved quantity 83 Robert McLeod Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Gaussian design Gaussian design aka thick lenses Thick lenses are fully characterized by three quantities 1 Effective focal length F or power 1FClgt 2 Front principal plane P 3 Rear principal plane P C P lt F In the paraxial limit all rays exiting a compound lens Will appear to have encountered a single plane P with power Ditto for reverse travelling rays but in general P is not at P Robert McLeod Mouroulis amp Macdonald 4 4 97 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Gaussian design Understanding principal planes Overlay previous two drawings and reverse the direction of the ray in the bottom picture if this bothers you hold on it will make sense in a moment Note that incident rays 1 and 2 appear to converge towards a virtual object at P Note also that exiting rays 1 and 2 emanate from a virtual image at P And finally note that the object and image height are equal l k 2 V M y M N P P 4 F F gt Therefore principal planes are the conjugates of unit magni cation Robert McLeod 98 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Gaussian design 80 why do we care Use of principal planes Since the principal planes are the conjugates of unit magnification we can replace any compound lens element with just P and P Rays that hit P must be imaged with unit transverse magnification and unit angular magnification to P In other words they teleport from P to P Ditto on the reverse trip All the element power CI is applied on the second plane P if going forward P if going backwards Note that all of our paraXial lens equations work with distances measured from the principal planes Robert McLeod 99 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Gaussian design Summary of 2 surface system lt d gt 4 I yl 5 1 P P l 142 5 y2 N T03 R 2 Td R 1 T a 21t 1 2 d 1 2 1 t dt1 1 1t 2 1t1t 1 21 aim 1 2 1 d 2 r1d 1 2 1 2 N q 2 q d Power 21 1 2 1 2 aka Gullstrand s eq N 11 M gt M Magnification 1 IICD 61 N 12 0 Z I th CW1 1 Conjugate condition N22 2 0 2 tl p F 61 2 1 Front focal plane tfbfp F l d 1 Back focal plane M 1 W F 5 0r 1 Z q Front principal plane Robert McLeod 100 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Gaussian design Nodal points A ray crossing the axis at the first nodal point emerges from the second nodal points parallel to itself Obviously for lenses with the same index in object and image space the nodal points are at the intersection of the principal planes and the axis This is not true in general e g the eye i Robert McLeod 101 Design oi ideal imaging systems with geometrical optics G au ssi an design 20 om lens Gaussian optics design example O 0 Fl Back focal plane 1 1 75 F 3ioem 545 Lens 1 f1100 Lens 2 f230 Location of P F Location of P 39BFP location held constant at 0 39Distance from P to BFP F 39Note that P P reversed from single lens case 39Ray traces of 2lens system solid and Gaussian system dashed agree 1004 50 7L 100 30 Robert McLeod O O lf 0 39 l 100 150 71 1 2301 80 60 40 20 ECE 5616 QB System Design 102 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows Optics of finite size Introduction Up to now all optics have been infinite in transverse extent Now we ll change that Types of apertures edges of lenses intermediate apertures stops Two primary questions to answer What is the angular extent NA of the light that can get through the system Aperture largest possible angle for object of zero height Depends on where the object is located What is the lar est object that can get throu h the system ield At the edge of the fiel the angular transmittance is one half of the on axis value Will find each of these with two particular rays one for each of above Will find two specific stops one of which limits aperture and one which limits field The conjugates to these stops in object and image space are important and get their own names Finally this will allow us to understand the total power efficiency of the system Robert McLeod 103 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows The aperture stop and the paraxial marginal ray Aperture stop ParaXial marginal ray I PMR a N N J I J D l A diaphragm that is n0t the aperture quot stop Launch an axial ray the paraxial marginal ray from the object Increase the ray angle until it just hits some aperture This aperture is the aperture stop The sin of a is the numerical aperture The aperture stop determines the system resolution light transmission efficiency and the depth of fieldfocus Robert McLeod 104 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows Pupils The images of the aperture stop Exit pupil I I Aperture stop and entrance pupil Paraxial marginal ray PMR O 06 n J lt A diaphragm that is n0t the aperture stop The entrance exit pupil is the image of the aperture stop in object image space Axial rays at the object image appear to enter exit the system entrance exit pupil When you look into a camera lens it is the pupil you see the image of the stop Both pupils and the aperture stop are conjugates Robert McLeod O Shea 24 MampM chapter 5 105 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows Windows Images of the field stop Aperture stop Field stop Launch an axial ray the chief ray from the aperture stop Increase the ray angle until it just hits some aperture This aperture is the eld stop The angle of the chief ray in object space is the angular eld of View The height of the chief ray at the object is the eld height The chief ray determines the spatial extent of the object which can be Viewed When that object is very far away it is convenient to use an angular field of View When the field stop is not conjugate to the object vignetting occurs cutting off half the light at the edge of the field Robert McLeod 106 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows Field stops and windows Definitions I I Aperture stop Entrance Window Field stop M amp Exit Window Chief ray MampM refer to the chief ray as the paraXial pupil ray PPR The entrance exit window is the image of the field stop in object image space Robert McLeod 107 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows Numerical aperture The measure of angular bandwidth N A E n Sin a Definition of numerical aperture Note 1nclus1on of n 1nth1s expression 192 ParaXial approximation Definition of F number ParaXial approximation 1 r0 06 z AF Radius of Airy disk NA is the conserved quantity in Snell s Law because it represents the transverse periodicity of the wave 2 2 kx 27 fx fnsinaz NA Therefore NAxto equals the largest spatial frequency that can be transmitted by the system Note that NA is a property of cones of light not lenses Robert McLeod 108 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows Effective F Lens speed depends on its use Infinite conjugate condition I 1 F 2NA 0c f D 39D Finite conjugate condition szng d li MF 7 D F0LM lt t quot t gt D D 1 F 01 Robert McLeod 109 Design of ideal imaging systems with geometrical optics ECE 5616 GE System Design Finite optics stops pupils and windows Depth of focus Dependence on F Your detection system has a finite resolution of interest eg a digital pixel size p Exit pupil Limited object DOF 1 t u 5 W39 M r4 Fm D 1 to In nity N wncm hm 4 lw wxh 7 u u v e v Source Wihpedia p From geometry F F or Fi de endin p p g NA on which space we re in 2 4iF 12 2 If r0 N A Airy disk diameter First eq is detector limited second is diffraction limited Thus as we increase the power of the system NA increases the depth of field decreases linearly for a fixed resolvable spot size p or quadratically for the diffraction limit r0 Robert McLeod MampM 513 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows Hyperfocal distance Important for fixedfocus systems 8 O a 8 ha E E 3 Entrance pupil S S E d o a LL Z Z f D lt gt 6z Note how each focus falls within Depth of Field the acceptable blur p The object distance which is perfectly in focus on the detector is defines the nominal system focus plane When this plane is positioned such that the far point goes to infinity then the system is in the hyperfocal condition and the nominal focus plane is called the hyperfocal distance This is very handy for fixed focus cameras you can take a portrait shot if the person is beyond the near point out to a landscape and not notice the defocus Robert McLeod 1 1 1 Design of ideal imaging systems with geometrical optics Finite optics stops pupils and windows Vignetting Losing light apertures or stops Take a cross section through the optical system at the plane of any aperture Launch a bundle of rays off axis and see how they get through this aperture UnVignetted a 2 iyl lyl Fully Vignetted a S iii iyi Robert McLeod ECE 5616 OE System Design 39Example of Vignetting Ray bundle of radius y Centr ray heighty WAperture radius 2 a Ray bundle of radius y All ray height r Aperture radius 2 a 112 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows Pupil matching Offaxis imaging amp vignetting Exit pupil of telescope image of AS in image space matches entrance pupil of following instrument eye Good place for afield stop if the object is at in nity Note that the entrance pupil in the system above will appear to be ellipsoidal for offaxis points Thus even in this welldesigned aberrationfree case offaxis points will not be identical on axis Vignetting Further we might find extreme rays at offaxis points are terminated This loses light bad but also loses rays at extreme angles which might limit aberrations good f1 JL Robert McLeod 1 13 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows Telecentricity Aperture stop 7 In a telecentric system either the EP or the XP is located at infinity The system shown above is doubly telecentric since both the EP and the XP are at infinity All doubly telecentric system are afocal When the stop is at the front focal plane just lens f2 above the XP is at infinity and slight motions of the image plane will not change the image height When the stop is at the back focal plane just lens fl above the EP is at infinity and small changes in the distance to the object will not change the height at the image plane Telecentricity is used in many metrology systems Robert McLeod 1 14 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Example Detailed example Holographic data storage reference y Reference Imaging Telescope Lens Holograp M aterial Galvo Mirror Linear CCD Correlation Imaging Lens We re going to design the imaging path Robert McLeod 1 15 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Example 1 to 1 imaging Aperture in Fourier plane Lens SLM A Aperture CCD Material 40 mm 20 mm 20 mm DSLleO mm W D Ap 2 mm DCCDle mm p Hm Lens10 mm f 20 mm Fourier diffraction calculation for a random pixel pattern 1 W39IS ww mmmWxgugnl nnfmmf 133 Robert McLeod 1 16 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Exampe Finding the aperture stop Trace the paraxial marginal rav PMR 0 1 2 3 SLM Lens Aperture CCD I yk1 1 t1 yk uk1 0 1 I V DSLM10 mm DEM 10 mm DAP 2 mm DCCDle mm p 10 um f 20 mm Aperture stop 7 1 2 4 2 Robert McLeod 1 17 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Exampe What does the aperture stop do Limits the NA of the system A yzz mm rAsz 1 mm X 40 mm What is the lens effective NA N A 2 D 2 Z 5 Z l f1 M 20 2 8 However we have stopped the system down to NA 2402120 xi 5 1 PixelsizelO m A M gt N W p 10 20 The aperture stop thus Determines the system field of View Controls the radiometric efficiency Limits the depth of focus Impacts the aberrations 5 Sets the diffractionlimited resolution PP PE Robert McLeod 1 18 Design of ideal imaging systems with geometrical 0 Example ptics ECE 5616 OE System Design Find the entrance pupil image of aperture stop in object space Aperture stop and exit pupil Entrance pupil gt 1 1 1 1 1 t1 00 t0 1 f 20 1 20 What s that mean If the entrance p upil is at infinity then every point on the object radiates into the same cone A k Robert McLeod 119 Design of ideal imaging systems with geometrical optics Find the field stop Trace the chief ray PPR Exampe Lens A Aperture CCD DSLleO mm p 10 mm V DEM 10 mm DAP 2 mm DCCDle mm f20mm Field stopr ECE 5616 OE System Design iiii ltliyk 0114C i Surf 0 1 1 lt 2 2 2 O 01 yr 04 04 PPR Y 5 5 5 Robert McLeod Entrance and exit windows at field stop since it is lens 120 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Exampe Required lens diameter And why windows should be at images Obiect Vignetted at edges Extreme mar inal Ia Note that the extreme marginal ray makes the same angle with the chief ray PPR as the paraxial marginal ray PMR makes with the axis Thus to fit the extreme marginal ray through an aperture we require Dzzqyiiyi22512 Now the camera is the field stop and the entrance window is the SLM The optical system now captures the same light from each SLM pixel and it can be adjusted for all pixels Via the aperture stop Independently we can adjust the field size Robert McLeod MampM 5 18 121 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Summary Thicken the lenses Gaussian design SLMltr 39 Aperture CCD DSLM10 mm p 10 um I Robert McLeod 122 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Finite optics stops pupils and windows Optical invariant aka Lagrange or Helmholtz invariant Write the paraXial refraction equations for the marginal ray PMR and chief or pupil ray PPR n39u39 nu y n39LT39 m7 With a bit of algebra niby 7y nu r y So H 1101 y 7 is conserved At the object or image of limited field diameter L y 0 i edge of field u 2 maximum ray angle L xi L xi Hob nuy 2 NA 6 z spots 2 r0 2 2 Thus we have found the information capacity of the optical system aka the spacebandwidth product 2 Nspots EHobj Robert McLeod 123 Design of ideal imaging systems with geometrical optics ECE 5616 013 system Design Summary Optical invariant for the holographic storage example N spots 2 27f ZyO NAO Definition of optical invariant 10 DSLM 2 2 T DSLM 9 Npix Stoppeddown NA So we have correctly designed this system to transmit the proper number of resolvable spots Robert McLeod 124 ECE 5515 OE System Design Paraxtal ray tracmg 01 Optical systems 7 ausstan beams Gaussian beams Solution of scalar paraxial wave equation Helmholtz equation is a Gaussian beam given by w e ziejkzejkzgf Wm E7 A0 0 g W Z z where wz Size Radius ofcurvature Gouy phase 2 z 2 z wow 1 Rltzgtz 70 mth 10 z Z0 3 Gaussian beam parameters 39 2 WEL 7 Zn 1 rr Wu 0 7 RE 71 7777 7 a za fee zo 73 73 72 71 o 1 2 3 zzU Note that Rz does not obey ray tracing sign convention Unfortunately there s no particularly good way to x this Saleh amp Telch chapter MdzMAppem39wt 2 Robert McLeod Paragtltiai ray tracing of Optical systems BCE 5616 OE System Desigi Aeaussian beams Gaussian beams Detailed view Real part of E vs radius andz rWn zzO At 2 20 10z0 10gt02 wz0 JEWO Rz0 2Z0 min value Any constant times wz is a ray path Note that the rays are converging and diverging spherical wave exceptnear the focus where they bendr Ergo rays do not always travel in straight lines 7 the region near the focus violates the slowly varying envelope approximation Conversion formulas 2 2 1 1 1 7r 2 w0 w 7 7i z 70 W i o zo 90 0 7r 0 l o 0 Robert McLeod Paraxia ray tracing of optical systems ECE 5616 OE System Design Gaussian beams Gaussian beam parameter qz The complete Gaussian beam expression normalized to intensity D is number of transverse dimensions 12 1 1 a 39 4r Define the complex radiusof curvatur Re 1mm What is qz qz 1 1 1 argq tan 1Z O ZZ0 z z2 z Maw T W Zgt Z j Z0 Z0 Z0 wO Can now write Gaussian above as 02 2 k p k 1 Zo que Ev jAO xEW 61Ze Note that phase of jqz is 4 Robert McLeod 86 Paraxia ray tracing of optical systems Gaussian beams How does q change with transfer and refraction Free space 5112Z1jzo 512 2Z2jzo 2511Z2Z1 Start with expression for qz s0q2q1Az Thin lens 1 1 j A St t 39th 39 f 1 ar W1 ex ress10n or z dz Rz if w2z p q Thin lens equation expressed as change in i i i curvature of wave R R f NOTE HOW GAUSSIAN BEAM SIGN CONVENTION HAS CHANGED THE SIGN i l i Apply to lq q q f L Solve for qf1 q Robert McLeod ECE 5616 OE System Design 87 Paraxia ray tracing of optical systems ECE 5616 OE System Design Gaussian beams ABCD approach to q Remember the ABCD matrices for thin lens refraction and free space transfer 1 O 1 t Rk Tk k k 1 0 1 and define the evolution equation for q Ag B q Cq D Check for free space 1t 2 q k t 061 1 Check for thin lens lq 0 q kq1 qf1 Which says rather remarkably that we can model the propagation of a Gaussian beam through a paraXial optical system using ray matrices But there s a better way to do so at least I like it s better Robert McLeod 88 Paraxia ray tracing of optical systems ECE 5616 OE System Design Gaussian beams Representation of Gaussian beams by complex rays 1 Define the following three rays Note their suggestive names and relationship to the Gaussian beam x2433quot Paraxial ray trajectory form quotvquot QZ W0 Wais AZ Z 90 1 D 662 M1010916X 60090 Chi f I e raV a ABCD vector form 5 C W a 0 f g 90 a g V g 0 a e A0 9 0 Define the complex ray trajectory This is Greynolds definition and yields the 1 Z J proper form of q Arnaud s de nition yields q You can then show that this ray contains qz FZ yA yo dFdz MA juQ Z6 0 jwo Ray heights over ray slopes Z Z0 Eg at z0 J Armand Applied Optics VQA N4 p 538 15 Feb 1985 A W Greynolds SPIE V 5601 p 33 1985 Robert McLeod M ampM A25 89 0 Paraxia ray tracing of optical systems ECE 5616 OE System Design Gaussian beams Representation of Gaussian beams by complex rays 2 First we note nl39 I Lagrange invariant Nspots 1 By brute for tracing of the rays we can find the following Gaussian parameters based on the two rays at that point yQuA Mun 90 J u 22 Which gives all other beam parameters wz1 y z yz le field radius at thisz We could use these two and the expressions for the Gaussian beam parameters to generate the complete Gaussian but this would be a bit tedious A more elegant way is to use the complex ray formalism Fz QZ dFdz yA ZMA jyo 549 A juo At plane z 750 Which apart from the onaXis phase kOS gives the full Gaussian beam at this plane z R Herloski S Marshall R Autos Applied Optics V22 N8 p 1168 Robert McLeod 15 Apr 1983 90 Paraxia ray tracing of optical systems ECE 5616 OE System Design Gaussian beams Representation of Gaussian beams by complex rays 3 OnaXis examples 1A1HmW0104f500Hm lf lfsystem 0 200 400 600 800 1000 z W O 500 1000 1500 2000 2 ml 3 A 1 am W0 2 f 32 f system O 500 1000 1500 2000 2500 2 ml Notes In 1 second waist is at Fourier plane as expected In 2 second waist occurs before image plane as expected In 3 as distance to lens increases waist moves to paraxial image plane Robert McLeod 91 Paraxia ray tracing of optical systems ECE 5616 OE System Design Gaussian beams Representation of Gaussian beams by complex rays 4 Offaxis examples 1A1HmW0104f500Hm lf lfsystem 30 20 10 mi 0 gt710 720 730 0 200 400 600 800 1000 z nml 0 500 1000 1500 2000 ZlumLquot 3 A 1 Mn W01 0quotf 500 f 32 f system 0 500 1000 1500 2000 2 ml Notes 1111 waist is centered at zero as expected of FT geometry In 2 image is at 10 um expected from Ml This type of problem is not possible with the ABCD formalism Robert McLeod 92 Paraxia ray tracing of optical systems ECE 5616 OE System Design Gaussian beams Do Gaussian beams obey paraxial imaging 13 f10 WZ n M l imaging MZ Object at waist mz t 40 30 720 10 Real object Answer E The image is also a Gaussian E field distribution in amplitude and any point on the object down from the peak by some value say le for the point wz will image to the point on the image down from the peak by the same value Shown above only for real objects conjugate to real images t 2 f Robert McLeod 93 Paraxia ray tracing of optical systems ECE 5616 OE System Design Gaussian beams Do Gaussian beams obey paraxial imaging 23 How about for Virtual objects M Virtual object fl 0 4 m m I I nun19 Answer Still works Robert McLeod 94 Paraxia ray tracing of optical systems ECE 5616 OE System Design Gaussian beams Do Gaussian beams obey paraxial imaging 33 How about for Virtual images M 75 f10 5 wz x Virtualimage xa x Real Ac x msg object 9z lt1 xx l 775 7 Conclusion All parts of the object Gaussian image correctly to the appropriate parts of the image Gaussian including both real and Virtual objects and images Corollary If you apply paraxial imaging to the object Gaussian over all z you generate the image Gaussian over all z Gaussian beams obey paraxial imaging exactly Robert McLeod 95 Design of ideal imaging systems with geometrical optics Gaussian beam propagation Design example Colimator lens 0 O 2L AB CD from start to center M TLR Td 1 L Ld1 L 1 d L Ld1 L jz01 L 1 d j zo W 2Z 1 d 2 q at center starting with q j z0 ReqL 0 gt L Where is waist Z0 Ll 1m What is new Rayleigh range J ZO NEW Robert McLeod ECE 5616 OE System Design 96
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