7- Sectio 7 Gaussia eductio
Paraxial aytrace Equatios eractio occurs at a iterace betwee two optical spaces. The traser distace t' allows the ray height y' to be determied at ay plae withi a optical space (icludig virtual segmets). u t C eractio: u u y y u u P P y u t y 7- Traser: y ytu t y yu y y
Gaussia eductio Gaussia reductio is the process that combies multiple compoets two at a time ito a sigle equivalet system. The Gaussia properties (power, ocal legths, ad the locatio o the cardial poits) are determied. Two compoet system System Power: Trace a ray parallel to the optical axis i object space. This ray must go through the rear ocal poit o the system. Paraxial raytrace: eractio Traser u u y y y y u t y y 7-3 u = u = 0 P P = = = = y P u = u u = u y P P 3 t = t Deie the system power by applyig the reractio equatio to the system: y 0 y
Two Compoet System System Power Trace the ray: u = u = 0 P P = = = = y P u = u u = u y t = t yy y y y y y P yy y y y P 3 t y 7-4 System power:
Two Compoet System ear Cardial Poits The system rear pricipal plae is the plae o uit system magiicatio. u = u = 0 P P = = = = y P y y d u d u d y y d t u = u u = u y t = t Note that the shit d' o the system rear pricipal plae P' rom the rear pricipal plae o the secod elemet P occurs i the system image space '. P d P P P 3 y u y y y 7-5 y P d I P is the rear vertex o the system, the the distace P is the BD.
Two Compoet System rot Properties epeat the process to determie the rot cardial poits. Start with a ray at the system rot ocal poit. It will emerge rom the system parallel to the optical axis. y u = u y P P P P P = = = = u u u = u = 0 t = t System: y 0 y 3 Work the ray backwards rom image space through the system. y 0 y y y y y y y y y y y y y t 7-6 System power: same result as or orward ray.
Two Compoet System rot Cardial Poits d y d u y u = u y P P P P P = = = = y u u=u u = u = 0 P d y y d t d t = t Note that the shit d o the system rot pricipal plae P rom the rot pricipal plae o the irst elemet P occurs i the system object space. 3 P y u y y P d y y I P is the rot vertex o the system, the the distace P is the D. 7-7
Gaussia eductio - Summary d = t P P P t d d P t P d t P = 3 7-8 E P ad P' are the plaes o uit system magiicatio (eective reractio or the system). d is the shit i object space o the rot system pricipal plae P rom the rot pricipal plae o the irst system P. d' is the shit i image space o the rear system pricipal plae Prom the rear pricipal plae o the secod system P. t is the directed distace i the itermediate optical space rom the rear pricipal plae o the irst system P to the rot pricipal plae o the secod system P. Both o these pricipal plaes must be i the same optical space. ollowig reductio, the two origial elemets ad the itermediate optical space are ot eeded or used.
Vertex Distaces The surace vertices are the mechaical datums i a system ad are ote the reerece locatios or the cardial poits. Back ocal distace BD: BD d rot ocal distace D: D d Object ad image vertex distaces are determied usig the Gaussia distaces: P d V D BD V d P 7-9 s d s d s V d P P d V s
The image part with relatioship ID rid8 was ot oud i the ile. Thick Les i Air A thick les is the combiatio o two reractig suraces. C C C 3 C C t C C CC t / t = V = C C d P P t d V t dvp t d VP 3 = = BD 7-0 BD d 3 PPt d dt PP The odal poits are located at the respective pricipal plaes.
The image part with relatioship ID rid was ot oud i the ile. Thi Les i Air t 0 0 C C C C =BD 7- E d d 0 BD The pricipal plaes ad odal poits are located at the les.
The image part with relatioship ID rid6 was ot oud i the ile. Two Separated Thi Leses i Air t t d t d t P P d d BD t = 7- BD d PP t d d t The odal poits are coicidet with the pricipal plaes.
Gaussia eductio Example Two Separated Thi Leses i Air Two 50 mm ocal legth leses are separated by 5 mm. t P t 5mm d V = BD 0.0 mm 0.0mm 0.03mm d t 5mm - 7-3 0.0 0.0 0.0 5 mm mm mm mm d6.667mm 0.03mm BD d 33.333mm BD 6.667mm
Diopters Les power is ote quoted i diopters D. Uits are m D D E - (i m ) ( i m) E With closely spaced thi leses, the total power is approximately the sum o the powers o the idividual leses. ocal legths do ot add. 7-4
Multi-Elemet eductio Multiple elemet systems are reduced two elemets at a time. A sigle system power ad pair o pricipal plaes results. Give these quatities, the ocal legths ad other cardial poits ca be oud. There are several reductio strategies possible or multiple elemets or suraces. 3 4 () (34) (34) 3 4 () 3 4 (3) 4 (34) 7-5 The system pricipal plaes are usually measured relative to the rot ad rear vertices o the systems: - The system rot pricipal plae is located relative to the rot pricipal plae o the irst surace or elemet. - The system rear pricipal plae is located relative to the rear pricipal plae o the last surace or elemet.
eductio i Pairs 3 4 () (34) (34) C C C 3 C4 V 3 3 4 t t t 3 3 4 V P P P P P 3 P P 3 4 P 4 t t t 3 d d d 34 d34 34 P P P 34 P 34 4 7-6 ttd d 34 d d d 34 ddd 34 34 t 3 d34 d 34 P P P 4 P d d
eductio Oe at a Time 3 4 () 3 4 (3) 4 (34) 3 3 4 4 3 4 P P P P P 3 P 3 P 4 P 4 t t t 3 d d P P t 3 d3 d 3 3 P 3 P 3 P 4 P 4 t 3 4 7-7 P P 3 3 P 4 P 4 ttd ttd 3 3 3 d d d d d d 3 34 34 d 34 P P t 3 4 d 34 P 4 P d d
Gaussia eductio Example Cemeted Doublet.0.57.649.0 C C C 3 V t t 73.8950 5.7840 6.5 3 C.03537 C.0930 C.006647 3.00700.0055.00400 3 3 V 4 7-8 t0.5 t 6.9 t 4.0 t 3.43
Gaussia eductio Example Cotiued irst, reduce the irst two suraces:.00457 3.86 d 0.60 d 3 7.48 P d V P V 3 d t V 7-9 t At this poit, the irst two suraces are represeted by ad the pricipal plaes ad P. t td.48 t 3.03 3 P
Gaussia eductio Example Cotiued Add the third surace: 3 3.00833 3 d 3 3 6.6 d 3 3 7.5 P V d P Object space Image space P 3 P d V 7-0 d t d 3 d 3 d d d 3 3.40 d 3 7.5.008333 0.0 0.0 E
Gaussia eductio Example Summary d d d 3 3.40 d 3 7.5.008333 0.0 0.0 E V P P D PP d d V BD 7- V P d 7.6 V P 3 d.85 rot ocal Distace D Back ocal Distace BD Pricipal plae separatio: PPVVd dttd d PP 4.95
7- eal Les to Thi Les Model
Sigle electig Surace Cosider a sigle relectig surace with a radius o curvature o. The rays propagate i a idex o reractio o. The agles o icidece ad reractio (I ad I') are measured with respect to the surace ormal. The ray agles U ad U', as well as the elevatio agle A o the surace ormal at the ray itersectio are measured with respect to the optical axis. The usual sig covetios apply. I I' electig Surace 7-3 O U V y U' O' A CC Curvature C Sag = /C Ceter o Curvature
Sigle electig Surace A ' U U O Curvature C I' I V y Sag U' O' electig Surace A = /C CC Curvature C Ceter o Curvature 7-4 elatig the agles at the ray itersectio with the surace: U A I I U A I U A Apply the Law o electio: I I U AU A
Sigle electig Surace ad the Law o electio II U AU A si U AsiU A siucos AcosUsi AsiUcos AcosUsi A si A si A siucosu siu cosu cos A cos A siucosuta AsiU cosu ta A Approximatio #: cosu cosu 7-5 Approximatio # implies that magitude o the ray agle is approximately costat. U U siu siu ta A cosu ta A cosu tauta AtaU ta A U' U
Paraxial Agles tauta AtaU ta A tautau ta A eormulate i terms o the paraxial agles or ray slopes: u tau u tau ta A uu ta A y Sag Approximatio #: Sag y y uu uuyc This is the Paraxial electio Equatio. V y Sag A = /C CC 7-6 Approximatio # implies that the sag o the surace at the ray itersectio is much less tha the radius o curvature o the surace.
electio ad eractio eractio: u u y u y C i uu y uyc uuyc electio Equals eractio with ELECTION Note that a relector with a positive curvature has a egative power. C u u y 7-7 C C electio: E C E C The rot ad rear ocal legths are equal to hal the radius o curvature.
Object ad Image distaces or a Sigle electig Surace The object ad image distaces ( ad ') are also both measured rom the surace vertex. O U I I' V y ' Sag electig Surace U' O' A CC 7-8 Approximatio #3: The object ad image distaces are much greater tha the sag o the surace at the ray itersectio. Sag Sag y y u tau Sag y y u tau Sag
Surace Vertex Plae ad Pricipal Plaes The same set o approximatios hold or paraxial aalysis o a relectig surace as or a reractio surace: The surace sag is igored ad paraxial relectio occurs at the surace vertex. The ray bedig at each surace is small. By igorig the surace sag i paraxial optics, the plaes o eective reractio or the sigle relectig surace are located at the surace vertex plae V. The rot ad ear Pricipal Plaes (P ad P') o the surace are both located at the surace. The odal poits o a relectig surace are located at its ceter o curvature as a ray perpedicular to the surace is relected back o itsel. 7-9 Vertex Plae electig Surace V P P' CC N, N'
Traser Ater electio Traser ater relectio works exactly the same as or a reractive system, except that the distace to the ext surace (to the let) is egative. y t The traser equatio is idepedet o the directio o traser. u y t 0 y y u t tu yy y y tu y y Ater relectio, the sigs o ad are opposite those o the correspodig u ad t. A drawig doe i reduced distaces ad optical agles will uold the mirror system ad show a thi les equivalet system. 7-30 Sig covetios ad relectio: - Use directed distaces as deied by the usual sig covetio. A distace to the let is egative, ad a distace to the right is positive. - The sigs o all idices o reractio ollowig a relectio are reversed.
Optical Suraces V > 0 > 0 > CC =- < 0 > 0 The rot ad rear pricipal plaes o a optical surace are coicidet ad located at the surace vertex V. Both odal poits o a sigle reractive or relective surace are located at the ceter o curvature o the surace. ( ) C E ( ) E : E CC C E V 7-3 A relective surace is a special case with C E C
eractive ad elective Suraces Power o a reractive surace: Power o a relective surace: ( ) C Assume ad.5 (0.5) C C / ( ) C C Assume 7-3 C or the same optical power, a relective surace requires approximately oe quarter o the curvature o a reractive surace. However the sigs o the powers are opposite. This is oe advatage o usig relective suraces.
Optical Suraces Cardial poits ( ) C Positive eractig: 0 C 0 E ( ) C E P, P at surace vertex N, N at ceter o curvature PP N,N CC 7-33 0 C 0 N,N CC PP
Optical Suraces Cardial poits ( ) C Negative eractig: 0 C 0 E ( ) C E N,N CC P, P at surace vertex N, N at ceter o curvature PP 7-34 0 C 0 PP N,N CC
Optical Suraces Cardial poits C Positive electig: Negative electig: 0 C 0 0 0 C 0 0 P, P at surace vertex C N, N at ceter o curvature E =- =- N,N CC PP PP N,N CC 7-35 A cocave mirror has a positive power ad ocal legth, but egative rot ad rear ocal legths. A covex mirror has a egative power ad ocal legth, but positive rot ad rear ocal legths.
eal ad Virtual Images eal images ca be projected ad made visible o a scree; virtual images caot. eal images the actual rays i image space head towards the image. Virtual the actual rays i image space head away rom the image. The rays must be projected backwards to id the image (virtual ray segmets). 7-36
Les Bedig The power o a thi les is proportioal to the dierece i the surace curvatures: C C Eve or a thick elemet, dieret shape leses ca be used to get the same power o ocal legth. The locatios o the pricipal plaes shit. P P d = 0 d = 0 7-37 P P or a relatively thi les, the pricipal plae separatio is idepedet o the les bedig.
System Desig Usig Thi Leses ) Obtai the thi les solutio to the problem: ) Iclude the pricipal plae separatios o real elemets: P P P P 7-38 3) Locate the vertices o the real compoets: V V V V P P P P The vertices ad vertex-to-vertex separatios are the mechaical datums or the system.
Gaussia Imagery ad Gaussia eductio The utility o Gaussia optics ad Gaussia reductio is that the imagig properties o ay combiatio o optical elemets ca be represeted by a system power or ocal legth, a pair o pricipal plaes ad a pair o ocal poits. I iitial desig, the P-P' separatio is ote igored (i.e. a thi les model). D V V P P V V BD 7-39
7-40 Methods o System Aalysis System Speciicatio adii, Idices, Thickesses, Spacigs, Elemet ocal Legths, etc. Imagig Properties Object ad Iage Locatios, Magiicatio, etc. aytrace aytrace aytrace Gaussia eductio Gaussia System System Cardial Poits Gaussia Imagery
Thi Les Desig Overall Object-to-Image Distace h ' L h m E h m m E m E h' 7-4 m L m m E E L m m E eal object ad real image: Miimum object to image distace: m 0 E E m L4 E
7-4 eciprocal Magiicatios Overall object-to-image distace: L m or each L, there are two possible magiicatios ad cojugates: eciprocal magiicatios. m m E m L E m m m m
Magiicatio Properties The Gaussia Magiicatio may also be determied rom the object ad image ray agles. h hp hp m u u h h u P h P hp hp u u P u h 7-43 hp hp h h P m h h P m u u This agle relatioship holds or all rays passig through o-axis cojugate poits.
Cardial Poits Example The power ad the relative locatios o the cardial poits o a system completely deie the imagig mappig. Dieret combiatios o elemets ca produce iterestig situatios. As a example cosider this true : imagig system cosistig o cascaded - systems. A iverted itermediate image is ormed. P P' 7-44 - - Because the object ad image plaes are plaes o uit magiicatio, the system rot ad rear pricipal plaes are coicidet with the object ad image plaes. Where are the ocal poits?
Cardial Poits Example - Cotiued To id the rear ocal poit o the system, lauch a ray parallel to the axis: P - -3.5 The system rear ocal poit is to the let o the system rear pricipal plae, ad the system power is egative! This iiity ray diverges rom the rear pricipal plae. d ' -.5 P' 7-45 t 4 4 t 0.5 SYSTEM / d t 4 / I a simpliied Gaussia model that igores the P-P' separatio, this system looks just like a egative thi les: By symmetry, the rot ocal poit o the system is to the right o the system rot pricipal plae. P P'
Mii Qui Two 00 mm ocal legth thi leses are separated by 50 mm. What is the ocal legth o this combiatio o leses? [ ] a. 66.67 mm [ ] b. 00 mm [ ] c. 50 mm [ ] d. Iiity 7-46
The image part with relatioship ID rid0 was ot oud i the ile. Mii Qui Solutio Two 00 mm ocal legth thi leses are separated by 50 mm. What is the ocal legth o this combiatio o leses? [X] a. 66.67 mm [ ] b. 00 mm [ ] c. 50 mm [ ] d. Iiity 00mm 0.0mm t 7-47 t 50mm t 0.05mm 66.67mm