Section 6. Object-Image Relationships

Transcription

1 6-1 Section 6 Object-Image elationships

2 Object-Image elationships The purpose o this study is to examine the imaging properties o the general system that has been deined by its Gaussian properties and cardinal points. Dierent combinations o ront and rear ocal lengths can be studied: 0 0 ositive ocal System 0 0 Negative ocal System 0 0 ositive ocal System; Net elective Negative ocal System; Net elective

3 Object-Image elationships In Terms o the ront and ear ocal Lengths Newtonian Equations (Origins at, '): Gaussian Equations (Origins at, '): 1 m mm 1 2 m 1 1 1m m 1 m 6-3 mm 1 2 Aocal Systems: 2 2 m 1 1 n m m m 1 2 n

4 Object-Image Zones Newtonian Equations (Origins at, ') Case A: < 0 Object to the Let o m 1 m m 0 0 m0 0 m0 m 0 m mm

5 ositive ocal System Object to the Let o eal Object eal Image A B C D 0 ' 0 m ' 0 (Newtonian distances) A' B' C' D' 6-5 This representation hides the act that both the object and image spaces are separate and extend rom minus ininity to plus ininity. In addition, the physical relationship between the locations o the ront and ear rincipal lanes on the system coniguration (number and type o elements including relective, spacings and thicknesses, reractive indices, etc.). hysically, ' can be to the right o, to the let o or coincident with. It depends on the system coniguration.

6 ositive ocal System Object to the Let o eal Object eal Image A B C D 0 0 m (Newtonian distances) 6-6 ' ' A' B' C' D' Images are inverted Objects and images are in the same order

7 Object-Image Zones Newtonian Equations (Origins at, ') Case B: > 0 Object to the ight o m 1 m m 0 0 m0 0 m0 m 0 m mm

8 ositive ocal System Object between and eal Object Virtual Image 0 A B C 0 m0 ( m1) 0 0 ( ) (Newtonian distances) A' B' C' ' ' Images are erect and magniied Objects and images are in the same order

9 ositive ocal System Object to the ight o Virtual Object eal Image 0 m0 (0m1) 0 0 ( 0) (Newtonian distances) A B C ' A' B'C' ' Images are erect and miniied Objects and images are in the same order

10 ositive ocal System Zones A B C ' ' B' C' m 1 1m 0 A' m 0 Image Space Object Space

11 Negative ocal System Object to the Let o eal Object Virtual Image A B C D 0 m0 (0m1) 0 0 (0 ) (Newtonian distances) ' A' B' C' D' ' Images are erect and miniied Objects and images are in the same order

12 Negative ocal System Object Between and Virtual Object eal Image 0 A B 0 m0 ( m1) 0 0 ( ) (Newtonian distances) C ' ' A' B' C' Images are erect and magniied Objects and images are in the same order

13 Negative ocal System Object to the ight o Virtual Object Virtual Image 0 0 m (Newtonian distances) A B C D 6-13 A' B' C' D' ' ' Images are inverted Objects and images are in the same order

14 Negative ocal System Zones 6-14 A B C 0 0 ' ' C' A' B' Image Space Object Space m 0 0 m 1 m 1

15 ositive ocal System elective Object to the Let o eal Object eal Image A B C D 0 0 m (Newtonian distances) 6-15 D' C' B'A' ' ' Images are inverted Objects and images are in the opposite order

16 ositive ocal System elective Object between and eal Object Virtual Image 0 A B C 0 m0 ( m1) 0 0 ( ) (Newtonian distances) ' ' C' B' A' Images are erect and magniied Objects and images are in the opposite order

17 This image cannot currently be displayed. This image cannot currently be displayed. This image cannot currently be displayed. This image cannot currently be displayed. This image cannot currently be displayed. ositive ocal System elective Object to the ight o Virtual Object eal Image A B C 6-17 ' C'B' A' ' Images are erect and miniied Objects and images are in the opposite order

18 ositive ocal System elective Zones A B C ' ' A' C' B' Image Space Object Space m 0 0 m 1 m 1

19 Object-Image Zones Summary ositive ocal System A B B C m > 1 1 > m > 0 ositive ocal System elective A A C A A B C A B B A m > 1 1 > m > 0 C m < 0 0 < m < 1 m > 1 Negative ocal System m < 0 0 < m < 1 m > 1 B Negative ocal System - elective C B B A m < 0 C C 0 0 C m < eal Virtual The object-image ones show the general image properties as a unction o the object location relative to the cardinal points An object in Zone A will map to an image in Zone A, etc. All optical spaces extend rom to. A net relective system (an odd number o relections) inverts image space about.

20 Object Space/Image Space Mapping ocal Systems m 1 and m 2 are the lateral magniications or the two planes Newtonian Equations (distances measured rom, ') 1 m m mm 1 2 m m The magniication is proportional to the Newtonian image distance and inversely proportional to the Newtonian object distance When Δ is small, the longitudinal magniication is obtained m m m 1 2 m m 2 m m The image space spacing is proportional to the Newtonian image distance squared and inversely proportional to the Newtonian object distance squared.

21 Object Space/Image Space Mapping ositive ocal System A B C ' ' B' C' A'

22 5-22 Object Space/Image Space Mapping Negative ocal System 0 A B C 0 ' ' C' A' B'

23 Collinear Transormation A collinear transormation maps points to points, lines to lines, and planes to planes. The general mapping equations associated with a collinear transormation are: x y axbycd axbycd axbycd axbycd axbycd axbycd By applying the symmetries associated with a rotationally-symmetric system and the deinitions o the magniication and the cardinal points, all o the relationships o Gaussian imagery can be derived (or both ocal and aocal systems) rom these general mapping equations These derivations have been prepared by ro. oland Shack and are included as Appendix A to these notes.