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Section 6

Object-Image Relationships

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pObject-Image Relationships

The purpose of this study is to examine the imaging properties of the general system that has been defined by its Gaussian properties and cardinal points.

Different combinations of front and rear focal lengths can be studied:

6-2

0 0 Positive Focal System

0 0 Negative Focal System

0 0 Positive Focal System; Net Reflective

0 0 Negative Focal System; Net Reflective

F R

F R

F R

F R

f f

f f

f f

f f

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Object-Image Relationships – In Terms of the Front and Rear Focal Lengths

Newtonian Equations (Origins at F, F'):

1 2

1F F

F R

F F F R

R

F

z z mf m fz z f f

fz m mz f

Gaussian Equations (Origins at P, P'):

1 2

11 1

1

F R

F R R

F

R

F

z z mf m f

f f fz mz z z f

fz m mz f

Afocal Systems: 2 2

1 1

2 21 2

1 2

F

R

R R

F F

f fmf f

f fz nm m mz f f n

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Object-Image Zones

Newtonian Equations (Origins at F, F')Case A: zF < 0 Object to the Left of F

1

0

0

F

F

FF

F

zf m

fzm

fm

0 0 0 0

0 0

F F

F F

R R

f m f m

z zm mf f

0 0 0 0

0 0 0 0

0 0 0 0

R R R R

F F F F

R R R R

F F F F

f f f f

z z z z

f f f ff f f f

1 2

R

F

fz m mz f

0 0z zz z

0 0z z

z z

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Positive Focal System0 00 0

F

R F

f mf z

Object to the Left of F Real Object – Real Image

0

zz

(Newtonian distances)

0Fz

F PBA C D

F' A'P' C'B' D'

Ff

Fz

Fz

Rf

This representation hides the fact that both the object and image spaces are separate and extend from minus infinity to plus infinity.

In addition, the physical relationship between the locations of the Front and Rear Principal Planes on the system configuration (number and type of elements – including reflective, spacings and thicknesses, refractive indices, etc.). Physically, P' can be to the right of P, to the left of P or coincident with P. It depends on the system configuration.

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Positive Focal System0 00 0

F

R F

f mf z

0

zz

Images are invertedObjects and images are in the same order

(Newtonian distances)

0Fz Object to the Left of F Real Object – Real Image

F PBA C D

F' A'P' C'B' D'

Ff

Fz

Fz

Rf

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Object-Image Zones

Newtonian Equations (Origins at F, F')Case B: zF > 0 Object to the Right of F

1

0

0

F

F

FF

F

zf m

fzm

fm

0 0 0 0

0 0

F F

F F

R R

f m f m

z zm mf f

0 0 0 0

0 0 0 0

0 0 0 0

R R R R

F F F F

R R R R

F F F F

f f f f

z z z z

f f f ff f f f

1 2

R

F

fz m mz f

0 0z zz z

0 0z z

z z

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Positive Focal System0 0 ( 1)0 0 ( )

F

R F F R

f m mf z z f

Object between F and P Real Object – Virtual Image

0

zz

(Newtonian distances)

0 F Fz f

Images are erect and magnifiedObjects and images are in the same order

F P

BA C

F'A'

P'C'B'

Ff

Fz

Fz

Rf

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Positive Focal System0 0 (0 1)0 0 ( 0)

F

R F R F

f m mf z f z

Object to the Right of P Virtual Object – Real Image (Newtonian distances)

F Fz f

Images are erect and minifiedObjects and images are in the same order

0

zz

F PBA C

F'

A'

P'

C'B'

Ff

Fz

Fz

Rf

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Positive Focal System – Zones00

F

R

ff

F P

F'P'

A B C

A'C'B'

FfRf

m 01 m 0 m 1

Obj

ect S

pace

Imag

e S

pace

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Negative Focal System0 0 (0 1)0 0 (0 )

F

R F F R

f m mf z z f

Object to the Left of P Real Object – Virtual Image (Newtonian distances)

F Fz f

Images are erect and minifiedObjects and images are in the same order

0

zz

FPBA C D

F'A'

P'C'B' D'

Ff

Fz

Fz

Rf

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Negative Focal System0 0 ( 1)0 0 ( )

F

R F F R

f m mf z z f

Object Between P and F Virtual Object – Real Image (Newtonian distances)

0F Ff z

Images are erect and magnifiedObjects and images are in the same order

0

zz

F'A'

P'C'B'

FP

BA C

Ff

Fz

Fz

Rf

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Negative Focal System0 00 0

F

R F

f mf z

Object to the Right of FVirtual Object – Virtual Image (Newtonian distances)

0Fz

Images are invertedObjects and images are in the same order

0

zz

FPBA C D

F'A' P'C'B' D'

Ff

Fz

Fz

Rf

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Negative Focal System – Zones00

F

R

ff

FP

F' P'

A B C

A'C' B'

FfRf

m 0 0 m 1 m 1

Obj

ect S

pace

Imag

e S

pace

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Positive Focal System – Reflective 0 00 0

F

R F

f mf z

Object to the Left of FReal Object – Real Image

0

zz

(Newtonian distances)

0Fz

Images are invertedObjects and images are in the opposite order

F PBA C D

F'A' P'C' B'D'

Ff

Fz

Fz

Rf

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Positive Focal System – Reflective 0 0 ( 1)0 0 ( )

F

R F F R

f m mf z z f

Object between F and P Real Object – Virtual Image

0

zz

(Newtonian distances)

0 F Fz f

Images are erect and magnifiedObjects and images are in the opposite order

F P

BA C

Ff

Fz

Fz

Rf

F'A'

P'C' B'

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Positive Focal System – Reflective This image cannot currently be displayed.

Object to the Right of P Virtual Object – Real Image

This image cannot currently be displayed.

Images are erect and minifiedObjects and images are in the opposite order

This image cannot currently be displayed.

F PBA C

Ff

This image cannot currently be displayed.

This image cannot currently be displayed.

Rf

F'

A'

P'

C'B'

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Positive Focal System – Reflective – Zones 00

F

R

ff

F P

A B C

F' P'

A' C' B'

FfRf

m 0 0 m 1 m 1

Obj

ect S

pace

Imag

e S

pace

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Object-Image Zones Summary

Real

Virtual

A BF P

m < 0m > 1 1 > m > 0

C

FPB C A z

z

AFP

m < 0m > 1 1 > m > 0

B C

AB CP F

z

z

AFP

m < 0 m > 10 < m < 1

B C

PFz

z

A BC

A BF P

m < 0 m > 10 < m < 1

C

A BCPF

z

z

Positive Focal System

Positive Focal System – Reflective

Negative Focal System

Negative Focal System - Reflective

The object-image zones show the general image properties as a function of the object location relative to the cardinal points.

An object in Zone A will map to an image in Zone , etc. All optical spaces extend from .

A net reflective system (an odd number of reflections) inverts image space about .

to

P

A

0 0F Rf f

0 0F Rf f

0 0F Rf f

0 0F Rf f

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Object Space/Image Space Mapping – Focal Systems

m1 and m2 are the lateral magnifications for the two planes

When Δz is small, the longitudinal magnification is obtained

1 2m m m

2R

F

fm mf

Newtonian Equations (distances measured from F, F')

F

F

fmz

2

2 2R F R F

F F F

f f f fmf z z

The image space spacing is proportional to the Newtonian image distance squared and inversely proportional to the Newtonian object distance squared.

1F F

F R

z z mf m f

1 2R

F

fz m mz f

F

R

zmf

The magnification is proportional to the Newtonian image distance and inversely proportional to the Newtonian object distance.

2 2

2R F F

F R R F

f z zmf f f f

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Object Space/Image Space Mapping – Positive Focal System 00

F

R

ff

F P

A B C

F'P'

A'C'B'

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Object Space/Image Space Mapping– Negative Focal System 00

F

R

ff

FP

A B C

F' P'

C' A' B'

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Collinear Transformation

A collinear transformation maps points to points, lines to lines, and planes to planes.The general mapping equations associated with a collinear transformation are:

1 1 1 1

0 0 0 0

2 2 2 2

0 0 0 0

3 3 3 3

0 0 0 0

a x b y c z dxa x b y c z d

a x b y c z dya x b y c z d

a x b y c z dza x b y c z d

By applying the symmetries associated with a rotationally-symmetric system and the definitions of the magnification and the cardinal points, all of the relationships of Gaussian imagery can be derived (for both focal and afocal systems) from these general mapping equations.

These derivations have been prepared by Prof. Roland Shack and are included as Appendix A to these notes.