Section 9. Paraxial Raytracing

Transcription

1 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp 9- Section 9 Paraxial atracing

2 YNU atrace efraction (or reflection) occrs at an interface between two optical spaces. The transfer distance t' allows the ra height ' to be determined at an plane within an optical space (inclding virtal segments). efraction or eflection: Transfer: n n nc n n t t n This tpe of ratrace is called a YNU ratrace. All ras propagate from object space to image space. 9- OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp A reverse ratrace allows the ra properties to be determined in the optical space pstream of a known ra segment. A ra can then be worked back to its origins in object space. efraction or eflection (reverse): n n Transfer (reverse): t

3 Paraxial atrace Eqations - Sstem Paraxial refraction eqation: n n efraction at an optical sstem effectivel occrs at the principal planes of the sstem. The ra emerges from the rear principal plane at the same height, bt with a different angle. Transfer: t P P t f E 9-3 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp The transfer distance t allows the ra height to be determined at an plane within an optical space (inclding virtal segments). Paraxial refraction occrs at the vertex plane of a srface. The srface sag is ignored. For a sstem represented b a power and a pair of principal planes, paraxial refraction occrs at the principal planes.

4 Paraxial atrace Single Srface Paraxial refraction occrs at the vertex plane of the srface. The srface sag is ignored. The image location is fond b solving for a ra height of ero. h n t V CC t =t n n t n h 9-4 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp n n t n n t t t n n h / n t / n n m h n t n n / / A ratrace spacing is the distance from the crrent srface to the next srface.

5 Paraxial atrace Single Component (in air) The principal planes are the locations of effective refraction. h P P t t =t t h 9-5 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp t t t t h t m h t These relationships also appl to a thin lens in air.

6 General atrace Eqations Transfer: efract: n t j j j j n n j j j j j j n n n n3 nk j j n k j j j j j j j j efraction occrs at each srface. The amont of ra deviation depends on the srface power and the ra height. Transfer occrs between srfaces. The ra height change depends on the ra angle and the spacing between srfaces. The image location is fond b solving for a ra height of ero in image space. 9-6 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp h k 3 k k k k+ t t t t t t 3 k t k h k

7 Paraxial atrace Series of Srfaces h t n n n n n3 nk k 3 Transfer: efract: t t t t t 3 k t k j j j j n n j j j j j j n k k k t k k+ h 9-7 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp k t t t k k k k t k k k k n n n n n n n 3 3 n n n k k k k k k Image Location and Magnification: k k t k k h n m h n k k k

8 Paraxial atrace Series of Components (in air) h t P P P P 3 t t t t 3 The general ratrace eqations hold (in air): t P P 3 j j j j j j j j 3 3 t t 3 4 j 4 h j 9-8 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp Each element or component refracts the ra, and the principal planes are the locations of effective refraction. Transfer occrs between the rear principal plane of one component and the front principal plane of the next. Image location and magnification: 4 t t h m h 3

9 atrace Example Two Separated Thin Lenses in Air Two 5 mm focal length lenses are separated b 5 mm. A mm high object is 4 mm to the left of the first lens. h mm t 4mm t t 5mm t t? 3. mm 3 - h? 9-9 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp.(Arbitrar) t 4. mm. t 4.5 mm..7 t 3 t t mm 3 h m h.7 h 4.9 mm

10 atrace Example (Contined) Two Separated Thin Lenses in Air A second ra can be traced to determine the image sie. h mm t 4mm. mm t t 5mm t t mm 3-9- h? OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp h..(arbitrar) t 4. mm.8 t 9.5 mm.8.37 t 3 h mm

11 atrace Example (Contined) Two Separated Thin Lenses in Air If the arbitrar initial angle of the second ra is chosen to be ero, the location of the rear focal point of the sstem can also be determined. h mm t 4mm t t 5mm t t mm 3 BFD F. mm - h? 9- OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp h. t. mm. t 5. mm..3 t 3 h mm BFD (transfer to = ): BFD 3 BFD 6.67 mm 3

12 Cardinal Points from a atrace ear Points The Gassian properties of an optical sstem can be determined sing a paraxial ratrace with particlar ras. ear cardinal points: Trace a ra parallel to the axis in object space. This ra mst go throgh the rear focal point of the sstem. The k th srface is the final srface in the sstem. Sstem: k k k n k n V P k d V BFD f k n F 9- OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp k n k f E f n BFD V F k k n k k dvp k k d BFD f As transfers: F: BFD P : k k kd k k k

13 atrace Example (Contined) Two Separated Thin Lenses in Air A ra from an axial object at infinit can be sed to determine the rear cardinal points. t t 5mm P d V f BFD F. mm OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp.. t.5 mm..3 BFD 3 BFD mm n f f mm.3 mm d VP mm or d BFD f BFD f mm

14 Cardinal Points from a atrace Front Points Trace a ra from the sstem front focal point that emerges parallel to the axis in image space. The reverse ratrace eqations are sed to work from image space back to object space. Sstem: k k n F FFD f F V d P k k V 9-4 k n k OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp n k k f E f F n n FFD VF d d FFD f k VP F F FFD FFD P : k d As transfers: :

15 Example Thick Lens in Air C =./mm = 5 mm C = -./mm = - mm t = mm n =.5 From Gassian optics (for comparison): =./mm =.5/mm =.467/mm f E = 68.6 mm n = n = n n 3 = C C V P P V d d t 9-5 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp ff 68.6 mm f 68.6mm d =.7 mm d' = mm PP 3.9mm There are several different spreadsheet forms that can be sed to facilitate the ratrace.

16 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp atrace Example Forward a 9-6 C t n Object Srface Space Srface Space Srface Space 3 - t/n n n Image Srface

17 atrace Example Forward a First, trace a ra parallel to the axis in object space to determine the rear focal point and rear principal plane. F' C t n - t/n n n Object Srface Space Srface Space Srface Space 3 a parallel to axis * + * = = *.9333 =? * = Image Srface 9-7 Solve to obtain = at F' OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp arbitraril chosen to eqal

18 atrace Example From the Trace of the Forward a VF n 3 VF mm / mm fe 68.6 mm f 68.6mm BFD V F mm d BFD f 4.54 mm f nf f nf n n 9-8 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp

19 atrace Example Front Properties Now, trace a ra from the front focal point that emerges parallel to the axis in image space to determine the front focal point and front principal plane. C t n - t/n F Object Srface Space Srface Space Srface Space 3?. FV Image Srface 9-9 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp n a b c a parallel to axis nafvb.ba c 6.667cb.5 c c.5 b.9667 a.467 FV 65.89

20 atrace Example everse a Use the reverse ratrace eqations. C t n - t/n n n F Object Srface Space Srface Space Srface Space 3? * = * = +. Image Srface 9- a parallel to axis OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp Solve for :

21 atrace Example From the Trace of the everse a FV FV mm n.467 fe ff.467 / mm 68.6 mm 68.6mm FFD VF FV mm d FFD f.7 mm F ff nf ff nf n n 9- OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp PPtd d mm

22 atrace Example For a Finite Object h OV s OV C t n - t/n n n * arbitrar h.5mm Object Srface Space Srface Space Srface Space 3 svi 98.3mm.5 m.5..*. * ? Image Srface Solve Image Location Image Sie Gassian check: d.7mm d 4.54mm.467 fe 68.6mm sd.7mm.8mm n. m.5 n.966 svi d98.3mm OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp

23 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp 9-3 YNU atrace Form Srface C t n - t/n n n n

24 atrace Example YNU atrace Form Srface 3 C. -. t /?/?/ /? n t/n /?/ 6.667?/ /? n n n Crvatres, powers and ra heights are associated with optical srfaces. Thicknesses, indices and angles are associated with optical spaces. 9-4 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp

25 Cemented Doblet C.3537 C.93 C n t.57 n t 4. Srface 3 4 C t.5 4.? n t/n n VF n' 9-5 VF BFD f E f. d BFD f 7.5 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp

26 9-6 Mirror Sstem Cassegrain Telescope mm t 8mm 5mm n n P V V F n n 3 n t WD d BFD C.5 C. f P n nc ( n3 n) C d F./ mm.4/ mm FFD f F Gassian edction: 8..4 (.)(.4)( )./ mm fe f 5mm f 5mm The front focal point and both principal planes are well in front of the sstem. F t 8mm 8mm n. 8 d 4mm..4 8 d 6mm. BFD f d mm FFD f d mm F WD BFD t mm OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp

27 Mirror Sstem Cassegrain Telescope atrace Srface 3 C t /? -8?/ n t/n /? 8?/. n n BFD V F mm../ mm f f 5mm E d BFD f 4mm WD BFD t mm FV mm FFD FV mm../ mm f 5mm f 5mm F d FFD f 6mm F E OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp

28 Paraxial atrace Thin Lens in Air The principal planes of a thin lens are both located in the plane of the lens. t 9-8 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp Power: f n n efraction: Transfer: t

29 a Deviation Thin Lens in Air The paraxial ra deviation introdced b a thin lens is independent of the object-image conjgates. emember that paraxial angles are actall ra slopes. It depends onl on the ra height at the lens and the lens power or focal length: 9-9 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp

30 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp 9-3 Thin Lens YU atrace Srface f - t t

31 Thin Lens Telephoto Lens f mm./ mm t 5mm f 75 mm.333/ mm t.333/ mm f f 3 mm E f 3mm f F 3mm P t < f f f d f BFD << f BFD F 9-3 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp d t mm 5 d t mm BFD f d 5mm FFD f d 5mm F

32 Thin Lens Telephoto Lens - atrace Srface f t /? 5?/ BFD V F 5mm / mm f f 3mm E d BFD f 5mm FV 5mm 9-3 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp FFD FV 5mm / mm f 3mm f 3mm F d FFD f mm F E

33 atrace Comments In a paraxial ratrace, t is the directed distance from the crrent srface to the next srface. As a reslt, real objects will sall have a positive distance to the first srface, as opposed to the tpical negative Gassian object distance. Srfaces are ratraced in optical order, not phsical order. All planes of interest in an optical space mst be analed before transferring to a reflective or refractive srface and entering the next optical space. Within an optical space, transfers move back or forth along the ra in that space withot changing the ra angle. eal and virtal segments of the space can be accessed OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp

34 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp YNU atrace Form 9-34 Srface C t n - t/n n n n

35 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp Thin Lens YU atrace Form 9-35 Srface f - t

36 Virtal Objects and atraces Consider an optical sstem or projector forming a real image. The ra heights and angles at the image are eas to determine. A second lens is now placed between the first lens and its image. The original image no longer exists, bt it now serves as a virtal object for the second lens. A final sstem image is formed b the second lens working with the first lens OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp Once again, determining the ra heights and angles in the sstem image space is straightforward. Transfer from the first lens to the second lens and refract. t

37 Virtal Objects and atraces A common sitation is to be given the sie and location of the virtal object for a lens or sstem withot an information abot the optical sstem sed to prodce it (the first lens on the previos slide). as mst be created corresponding to this intermediate image (the virtal object). These ras exist in the optical space corresponding to the virtal object, which is the object space of the optical sstem. Once the intermediate space ras are defined, these ras are transferred b t back to the entr vertex of the optical sstem. The ra heights and angles are now known at the first vertex of the optical sstem, and the are in the sstem object space. These constrcted ras can then be propagated throgh the sstem to the sstem image space. Pick two ras, one throgh the top of the virtal object, the other throgh the axial object point. The angles are arbitrar. Constrcted as in Object Space t h Virtal Object 9-37 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp

38 Virtal Objects and atraces When transferring back to the front vertex, these ras defining the virtal object are not refracted b the optical sstem. These ras are alread in object space. The negative thickness t transfers to the left along the virtal segments of the ras. egardless of the phsical order, ras are traced in optical order: from object space to image space. The thicknesses are the directed distances as defined b the sign conventions. Constrcted as in Object Space h Virtal Object 9-38 OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp t The transfer distance t is the directed distance from the virtal object to the front vertex of the sstem. This transfer distance is in object space and is associated with the index of refraction of object space. The powers and indices of the elements mst not be sed in transferring back to the front vertex of the sstem. If a sstem represented jst b its power and principal planes is to be analed, the transfer distance is the distance from the virtal object to the front principal plane of the sstem P.

39 Virtal Object Example A mm high virtal object is located 4 mm to the right of the first srface of the following mm focal length convex-plano thick lens: = 5.7 mm = Infinit t = mm n =.57 C =.934/mm C = Determine the image location and sie. Srface Obj Image C.934 t n ?. - t/n n n -4 -.* -. * -. * Arbitrar Lanch ras from the axial object location and from the top of the object at arbitrar angles. Transfer these ra heights to the first srface of the lens (t = -4 mm). There is no refraction associated with this transfer as these are object-space ras. Solve for the image location and the image height: The image is located.98 mm to the right of the rear srface of the lens. The image height is 7.43 mm. This is a virtal object prodcing a real image. OPTI-/ Geometrical and Instrmental Optics Copright 8 John E. Greivenkamp