Section 5. Gaussian Imagery
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1 OPTI-01/0 Geoetrical ad Istruetal Optics 5-1 Sectio 5 Gaussia Iagery
2 Iagig Paraxial optics provides a sipliied etodology to deterie ray pats troug optical systes. Usig tis etod, te iage locatio or a geeral syste ca be calculated relative to te Pricipal Plaes o te syste. Tis as also allowed te ocal Legt o a geeral syste to be deied i ters o te syste s overall reractive properties. However, te pysical locatios o te Pricipal Plaes or a give optical coiguratio still eed to be derived, so te aalysis i ot yet coplete. Te details o deteriig te syste ocal legt o a cobiatio o optical eleets is also ot clear. I additio, te iage sie is as o yet ukow. As will be deostrated later, paraxial raytrace ca be used to aswer tese questios. 5- OPTI-01/0 Geoetrical ad Istruetal Optics Gaussia optics provides a alterative etod o syste aalysis tat treats iagig as a appig ro object space ito iage space.
3 Gaussia Optics Gaussia optics treats iagig as a appig ro object space ito iage space. It is a special case o a colliear trasoratio applied to rotatioally syetric systes, ad it aps poits to poits, lies to lies ad plaes to plaes. Te correspodig object ad iage eleets are called cojugate eleets. Assuptios: Axial Syetry Meridioal Plae - cotais te axis o syetry - usually te y- plae Sig Covetios ad deiitios o reerece locatios will be applied as usual. Te trasverse agiicatio or lateral agiicatio is te ratio o te iage poit eigt ro te axis to te cojugate object poit eigt: 5-3 OPTI-01/0 Geoetrical ad Istruetal Optics Te udaetal priciple o geoetrical optics olds: ay tree-diesioal object scee is represeted by a collectio o idepedetly radiatig poit sources. ac poit source is idepedetly iaged troug te syste to its cojugate iage poit (perect iagery). Te tree-diesioal iage is te superpositio o all o te poit iages.
4 Gaussia Optics Teores Teores (all derive ro rotatioal syetry o te syste): - Plaes perpedicular to te axis i oe space are apped to plaes perpedicular to te axis i te oter space. - Lies parallel to te axis i oe space ap to cojugate lies i te oter space tat eiter itersect te axis at a coo poit (ocal syste), or are also parallel to te axis (aocal syste). ocal Syste: 5-4 OPTI-01/0 Geoetrical ad Istruetal Optics Aocal Syste: I a Gaussia or colliear appig, lies ust ap to lies ad are cojugate eleets.
5 Gaussia Optics Teores - Cotiued - Te trasverse agiicatio is costat i cojugate plaes perpedicular to te axis. I te agiicatio i a plae were ot costat, te iages o te grid lies would becoe curved or distorted. 5-5 OPTI-01/0 Geoetrical ad Istruetal Optics
6 Cardial Poits ad Plaes Te cardial poits ad plaes copletely describe te ocal appig. Tey are deied by speciic agiicatios: rot ocal poit/plae = ' ear ocal poit/plae = 0 P rot pricipal plae = 1 P' ear pricipal plae = 1 Cosider cojugate lie eleets i object ad iage space: H P H OPTI-01/0 Geoetrical ad Istruetal Optics P 0 H 1 H Te rot ad rear ocal legts ( ) are deied as te directed distaces ro te ad rot ad rear pricipal plaes to te respective ocal poits.
7 Cardial Poits ad Plaes Upried variables are i object space. Pried variables are i iage space. AP rot atipricipal plae/poit = -1 AP' ear atipricipal plae/poit = -1 N rot odal poit Agular Mag = 1 N' ear odal poit Agular Mag = 1 Te two odal poits o a syste are cojugate poits. θ N N θ 5-7 OPTI-01/0 Geoetrical ad Istruetal Optics or a ocal iagig syste, a object plae locatio is related to its cojugate iage plae locatio troug te trasverse agiicatio associated wit tose plaes. Newtoia equatios easure object ad iage distaces ro te ocal plaes. Gaussia equatios easure object ad iage distaces ro te pricipal plaes.
8 Locatig a Iage wit te Cardial Poits Te optical syste ca be represeted as a set o Pricipal Plaes ad a set o ocal Poits (wit te respective ocal legts). - A ray icidet o te rot pricipal plae will eerge ro te rear pricipal plae at te sae eigt. - A ray parallel to te optical axis i object space passes troug te rear ocal poit. - A ray troug te rot ocal poit eerges parallel to te optical axis. - Te itersectio o two rays deies a object or iage poit. P P 5-8 OPTI-01/0 Geoetrical ad Istruetal Optics Te Pricipal Plaes serve as te plaes o eective reractio betwee object space ad iage space.
9 OPTI-01/0 Geoetrical ad Istruetal Optics Locatig a Iage wit te Cardial Poits xaple 1 Positive Syste eal object to te let o te rot ocal poit 5-9 P P P P
10 Locatig a Iage wit te Cardial Poits xaple Positive Syste eal object betwee te rot ocal poit ad te rot pricipal plae P P 5-10 OPTI-01/0 Geoetrical ad Istruetal Optics P P Te two iage space rays diverge ad ave a virtual crossig. A elarged, erect virtual iage is produced. Te iage is i iage space.
11 Locatig a Iage wit te Cardial Poits xaple 3 Negative Syste eal object Note te locatios o te ocal Poits. Te rot ocal Poit ad rot Pricipal Plae P are bot i te syste object space. Siilarly, te ear ocal Poit ' ad ear Pricipal Plae P' are bot i te syste iage space. Te sae iage oratio rules apply. P P 5-11 OPTI-01/0 Geoetrical ad Istruetal Optics P P Oce agai, te two iage space rays diverge ad ave a virtual crossig. A iiied, erect virtual iage is produced. Te iage is i iage space.
12 Newtoia quatios Te Newtoia equatios caracterie tis Gaussia appig we te axial locatios o te cojugate object ad iage plaes are easured relative to te respective ocal Poits. By deiitio, te rot ad rear ocal legts cotiue to be easured relative to te pricipal plaes. Use siilar triagles. ta 1 1 P ta P 0 0 Object Distace = Iage Distace = 5-1 Magiicatio: - Idepedet o te object eigt - Proportioal to te iage distace - Iversely proportioal to te object distace OPTI-01/0 Geoetrical ad Istruetal Optics
13 Newtoia quatios Applied to a Syste Te Newtoia equatios caracterie tis Gaussia appig we te axial locatios o te cojugate object ad iage plaes are easured relative to te respective ocal Poits. P P 5-13 OPTI-01/0 Geoetrical ad Istruetal Optics I air: 1
14 Gaussia quatios Te Gaussia equatios describe te ocal appig we te respective Pricipal Plaes are te reereces or easurig te locatios o te cojugate object ad iage plaes. Use te sae siilar triagles. ta P P ta Object Distace = Iage Distace = OPTI-01/0 Geoetrical ad Istruetal Optics Add: atio: Magiicatio: - Idepedet o te object eigt - Proportioal to te ratio o te iage distace to te object distace /
15 Gaussia quatios Applied to a Syste Te Gaussia equatios describe te ocal appig we te respective Pricipal Plaes are te reereces or easurig te locatios o te cojugate object ad iage plaes P P OPTI-01/0 Geoetrical ad Istruetal Optics 1 1 / 1 / I air:
16 Cojugate plaes or = 1, 0 ad Use te Gaussia equatios. = 1 = Object plae located at P Iage plae located at P P ad P' are cojugate. Object plae located at Iage plae located at ' OPTI-01/0 Geoetrical ad Istruetal Optics ad ' are cojugate. = Object plae located at Iage plae located at ad are cojugate.
17 Sig Covetios evisited Newtoia quatios Derivatio Origial Coiguratio Object to te let o ;, ad are egative. 0 P New Coiguratio Object to te rigt o : ow oly is egative. ta 5-17 OPTI-01/0 Geoetrical ad Istruetal Optics P 0 ta Sae result!
18 Sig Covetios evisited Origial igure Object to te rigt o ; ow,, ad are egative. P 0 ta Sae result! 5-18 OPTI-01/0 Geoetrical ad Istruetal Optics As te object distace goes positive to place te object to te rigt o, te object eigt becoes egative to copesate i te equatios. Te et result is always te sae. Note tat i all o tese igures, te reerece locatio used to deie te quatities as ot caged. Set te equatios up to be cosistet wit te igure, te sig covetios will allow te equatio to be valid or dieret coiguratios.
19 Distaces Betwee Pairs o Cojugate Plaes Tickess Magiicatio Te tickess agiicatio relates te distaces betwee pairs o cojugate plaes. Use Newtoia equatios: P P OPTI-01/0 Geoetrical ad Istruetal Optics / 1 / / Te tickesses ad are idepedet o te origis used.
20 Tickess Magiicatio ad Logitudial Magiicatio 1 / 1 / 1 Te tickess agiicatio equatios are valid or widely separated plaes. Sice it is a dierece i positio, te result is idepedet o te coice o origis. As te plae separatio approaces ero, te local logitudial or axial agiicatio is obtaied. 5-0 OPTI-01/0 Geoetrical ad Istruetal Optics / li 0 / Sice varies wit positio, te logitudial agiicatio ad te tickess agiicatio are a uctio o ad.
21 Tickess Magiicatio ad Logitudial Magiicatio Syste Te tickess agiicatio relates te distaces betwee pairs o cojugate plaes. 1 1 P P I air: OPTI-01/0 Geoetrical ad Istruetal Optics Te tickess agiicatio equatios are valid or widely separated plaes. As te plae separatio approaces ero, te local logitudial or axial agiicatio is obtaied: li 0 1 I air: Sice varies wit positio, te logitudial agiicatio ad te tickess agiicatio are a uctio o ad.
22 Nodal Poits Two additioal cardial poits are te rot ad rear odal poits (N ad N') tat deie te locatio o uit agular agiicatio or a ocal syste. A ray passig troug oe odal poit o a syste is apped to a ray passig troug te oter odal poit avig te sae agle wit respect to te optical axis. Te odal poits are cojugate poits. P N PN P N PN ay 1 ay ays 1 ad ust be parallel i iage space, sice teir cojugate rays cross i te rot ocal plae. Te idicated triagles are ot oly siilar, but idetical. 5- OPTI-01/0 Geoetrical ad Istruetal Optics PN PN PN PN PN Usig te tickess agiicatio (ad te locatios o te pricipal plaes ad odal poits): N P PN N P PN P N 1 P 1 N N
23 Nodal Poits o a Syste PN PN N I te sae idex occurs i object space ad iage space: 5-3 OPTI-01/0 Geoetrical ad Istruetal Optics PN PN 0 N 1 Te odal poits are located at te respective pricipal plaes i te iage space idex o reractio equals te object space idex o reractio.
24 Origis at te Nodal Poits I te object ad iage locatios are easured relative to te Nodal poits, a iterestig ad iportat result is obtaied. Use te tickess agiicatio relatiosip: N N N N N N ad are te object ad iage positios easured ro soe origis ad are te odal poit positios easured ro te sae origis N 1 N N N 5-4 OPTI-01/0 Geoetrical ad Istruetal Optics Te agular subtese o a iage as see ro te rear odal poit equals te agular subtese o te object as see ro te rot odal poit. u N N N N u=u u u N N N N
25 Cardial Poits ad Plaes o a Syste Te syste ocal legt is Te object space idex o reractio is Te iage space idex o reractio is Optical Syste V P P V d d BD 5-5 OPTI-01/0 Geoetrical ad Istruetal Optics D Te rot ocal Distace D is te distace ro te syste rot vertex to te rot ocal poit Te Back ocal Distace BD is te distace ro te syste back vertex to te back ocal poit d is te sit o te syste rot Pricipal Plae P ro te syste rot vertex V d is te sit o te syste rear Pricipal Plae P ro te syste back vertex V
26 Nodal Slide We a les syste is i air, te odal poits coicide wit te pricipal poits. Te use o a odal slide allows te pricipal plaes ad te ocal legt to be experietally deteried. We te les is rotated about its rear odal poit, te rays will coverge to te sae poit. Te iage will ot ove eve toug te ray budle orig te iage is skewed, ad ' is sited to oe side. P,N P,N V 5-6 OPTI-01/0 Geoetrical ad Istruetal Optics P,N P,N By ivertig te les, te rot cardial poits (N,, P) ca also be located.
27 Nodal Slide Procedure P,N P,N d V BD P,N P,N Mout te les syste o a traslatio stage wic is o a rotatio stage. Positio te rear vertex over te rotatio axis. We properly positioed, te vertex will ot traslate we te les is rotated. Use colliated illuiatio. Use a icroscope (wit a icroeter) to easure te distace betwee te rear vertex V' ad te ocus (rear ocal poit '). Tis is te Back ocal Distace BD te distace ro te rear vertex to te rear ocal poit. Wile observig te iage, repositio te les wit te traslatio stage so tat te iage does ot traslate we te les is rotated. Te rear odal poit (ad te rear pricipal poit) are ow over te rotatio poit. Te aout te les was oved is te separatio d' betwee te rear vertex ad te rear pricipal plae. Te syste ocal legt is oud by BDd 5-7 OPTI-01/0 Geoetrical ad Istruetal Optics
28 Gaussia Properties o a Sigle eractig Surace Pricipal Plaes: Te rot ad rear pricipal plaes are coicidet ad located at te surace vertex V. A ray icidet o te surace at soe eigt will eerge ro tat surace at te sae eigt. Uit agiicatio occurs at te surace. Nodal Poits: Bot odal poits are located at te ceter o te curvature CC o te optical surace. A ray eadig towards te ceter o curvature is oral to te reractig surace ad is ot reracted. Te ray as te sae agle i object space ad iage space (agular agiicatio = 1), ad te ray s axial itercept at te CC deies bot odal poits. V N, N 5-8 CC OPTI-01/0 Geoetrical ad Istruetal Optics Surace Power ad ocal Poits: C Te rot ad rear ocal legts are easured ro te surace vertex.
29 Nodal Poits o a Sigle eractig Surace Veriy te Nodal Poit Pricipal Plae separatio: PN PN PN PN 1 PN PN C C 5-9 OPTI-01/0 Geoetrical ad Istruetal Optics Te pricipal plaes are at te surace vertex Te odal poits are at te surace ceter o curvature. Nodal poit agiicatio or a sigle reractig surace: N
30 Gaussia Properties o a eractig Surace Suary Power: ocal Legts: 1 C Pricipal plaes/poits located at te surace vertex. Nodal poits located at te ceter o curvature o te surace. N Te rot ad rear ocal legts are easured ro te surace vertex (pricipal plaes). V CC > 0 > 0 > 5-30 OPTI-01/0 Geoetrical ad Istruetal Optics Pricipal plae - Nodal poit separatio: 1 PN PN C Iagig: 1 Magiicatio: / /
31 educed Distace quivalece Cosider te exaple o a reractig surace ad its ti les equivalet. Bot ave te sae power. 1 u = 1 = 1 = / u=u Te reduced ocal legt o te surace equals te ocal legt o te ti les. I te ray agle or slope or te reractig surace is u', te te ray agle is 'u' or te ti les. A ray agle ultiplied by te reractive idex o its optical space is called a optical agle: u. Te paraxial raytrace equatio is i ters o optical agles ad te eective (or reduced) ocal legt o te syste: y y u u y u u / 5-31 OPTI-01/0 Geoetrical ad Istruetal Optics Te use o reduced distaces ad optical agles allows ay syste to be represeted as a air-equivalet syste coprised o ti leses. I te object is ot at iiity, te iage distace or te ti les becoes te reduced iage distace o te reractig surace.
32 Gaussia Iagery Te cardial poits, alog wit te associated ocal legts ad power, copletely speciy te appig ro object space ito iage space or a ocal syste. Gaussia iagery ais to reduce ay ocal iagig syste, regardless o te uber o suraces, to a sigle, uique set o cardial poits. 5-3 OPTI-01/0 Geoetrical ad Istruetal Optics P P P P
33 Iagig xaple 1 Positive ocal Syste Object: 00 to let o = 10 Use Newtoia equatios: OPTI-01/0 Geoetrical ad Istruetal Optics P P' ' Note: Te pysical separatio betwee P ad P' is ot kow.
34 Iagig xaple 1 Gaussia quatios Use Gaussia equatios: Distaces ro Pricipal plaes OPTI-01/0 Geoetrical ad Istruetal Optics 50 to te rigt o ' P P' '
35 Iagig xaple Sae Positive ocal Syste Object: 40 to rigt o = 10 Use Newtoia equatios: OPTI-01/0 Geoetrical ad Istruetal Optics P P' '
36 OPTI-01/0 Geoetrical ad Istruetal Optics xaple Suary 5-36 P P' B A ' A B or coveiece, te Pricipal Plaes are sow as coicidet.
37 5-37 Tickess ad Logitudial Magiicatio ocal Systes / or / ad are te lateral agiicatios or te two plaes Gaussia quatios (distaces easured ro P, P') / / 1 1 Newtoia quatios (distaces easured ro, ') 1/ / / / / / 1 OPTI-01/0 Geoetrical ad Istruetal Optics / / We Δ is sall, te logitudial agiicatio is obtaied 1 1 Gaussia: Newtoia: Te iage space spacig is iversely proportioal to te Newtoia object distace squared. / / / / 1
38 Iagig xaple 3 Logitudial Magiicatio Sae Positive ocal Syste Objects: 410 to let o 400 to let o Use Newtoia equatios: OPTI-01/0 Geoetrical ad Istruetal Optics xact tickess agiicatio:
39 Iagig xaple 3A Logitudial Magiicatio Sae Positive ocal Syste Objects closer to te syste: Use Newtoia equatios: 1 50 to let o 40 to let o OPTI-01/0 Geoetrical ad Istruetal Optics 49.4 xact tickess agiicatio:
40 Object Iage elatiosips Te geeral iagery relatiosips ca be writte eiter i ters o te rot ad rear ocal legts or i ters o te eective ocal legt ad idices o reractio. 1 or clarity, te expressios will explicitly ote te ocal legt as. As te course progresses, te subscript will be dropped, ad te ore coo expressio or te ocal legt will be used. Newtoia quatios (Origis at, '): 5-40 OPTI-01/0 Geoetrical ad Istruetal Optics 1
41 Object Iage elatiosips Page Gaussia quatios (Origis at P, P'): / / OPTI-01/0 Geoetrical ad Istruetal Optics Tickess ad Logitudial Magiicatio ocal Syste: Magiicatio o te Nodal Poits: N N
42 educed Distaces ad Optical Agles We te Gaussia iagery equatios are expressed i ters o te L or power ( or ), all o te axial distaces appear as a ratio o te pysical distace to te idex o reractio i te sae optical space. Tis ratio is called a reduced distace ad is usually deoted by a Greek letter. or exaple represets te reduced distace associated wit te tickess t: t Te L is te reduced ocal legt: it equals te reduced rear ocal legt or ius te reduced rot ocal legt. A ray agle ultiplied by te reractive idex o its optical space is called a optical agle: 5-4 OPTI-01/0 Geoetrical ad Istruetal Optics u
43 Iagig quatios i Air 1 Newtoia quatios (Origis at, '): 1 Gaussia quatios (Origis at P, P'): 1 1 Tickess ad Logitudial Magiicatio ocal Syste: OPTI-01/0 Geoetrical ad Istruetal Optics 1 Magiicatio o te Nodal Poits: Aocal Systes: N 1 1 1
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