J. H. Burge

3. Image motion due to optical element motion

• Tilt and decenter of optical components (lenses, mirrors, prisms) will cause motion of the image– Element drift causes pointing instability

•Affects boresight, alignment of co-pointed optical systems•Degrades performance for spectrographs

– Element vibration causes image jitter•Long exposures are blurred•Limit performance of laser projectors

Small motions, entire field shifts (all image points move the same)

Image shift has same effect as change of line of sight direction

(defined as where the system is looking)

J. H. Burge, “An easy way to relate optical element motion to system pointing stability,” in Current Developments in Lens Design and Optical Engineering VII, Proc. SPIE 6288 (2006).

J. H. Burge

Lens decenter

• All image points move together• Image motion is magnified

J. H. Burge

Effect for lens tilt

• Can use full principal plane relationships• Lens tilt often causes more aberrations than image motion

J. H. Burge

What happens when an optical element is moved?

To see image motion, follow the central ray

Generally, it changes in position and angle

y

s

Initial on-axis ray

y

s

Initial on-axis ray

Element motions : decenter : tiltCentral ray deviationy : lateral shift : change in angle

J. H. Burge

Lens motion

s

f

s

f

s

s

f

(Very small effect)

s

f

s

f

decentertilt

J. H. Burge

Mirror motion

s

s

f

s

s

f

s

s

f

2s

f

like lens = 2like flat mirror

J. H. Burge

Motion for a plane parallel plate

y

Plane parallel platethickness tindex n

1t ny

n

y

Plane parallel platethickness tindex n

1t ny

n

No change in angle

J. H. Burge

The Optical Invariant

The stop is not special. Any two independent rays can be used for this. The optical invariant will be maintained through the system

J. H. Burge

J. H. Burge

“Beam footprint”on element i

Di

Light from point on axis,

Bundle defined by aperture

Off-axis light is ignored

Element i

Nominal marginal rays at element i

ui = NAi

Perturbed central ray from element i

i iu

i iy y

yi

Image shift

NA and Fn based on system focus

“Beam footprint”on element i

Di

Light from point on axis,

Bundle defined by aperture

Off-axis light is ignored

Element i

Nominal marginal rays at element i

ui = NAi

Perturbed central ray from element i

i iu

i iy y

yi

Image shift

NA and Fn based on system focus

General expression for image motion

in i i i

NAF D y

NA

Fn final working f-number =

Di beam footprint for on-axis bundle

i = change in central ray angle due to motion of element i

1

2NA

J. H. Burge

Example for change in angle

n i iF D Image motion from change in ray angle

n

f

F D

For single lens, this is trivial

D

f = FnD

J. H. Burge

Effect of lens decenter

Decenter s causes angular change in central ray

Which causes image motion

Magnification of Image / lens motion

ii

s

f

n

i

i

Fs f

D

n i i n ii

sF D F D

f

Di is “Beam footprint” on element i

NA and Fn based on system focus

DiDi

J. H. Burge

Tilt causes angular change in central ray

Which causes image motion

“Lever arm” of 2 Fn Di ( obvious for case where mirror is the last element)

Example for mirror tilt

2 n i iF D

2i

2n i

n i

d

F D

F D

dFollow the central ray Small angle approx

Di is beam size at mirror

This is valid for any mirror!

J. H. Burge

Stationary point for finite conjugates

• Rotate about C, define system using principal planes

yC

c

d’

yC

c

d’

''stationary

fCP PP

d

For a thin lens, PP’ = 0, and the stationary point occurs at CP = 0, rotating about the principal points.

For an object at infinity, d’ = f , so CPstationary = -PP’, which means that the stationary point occurs at the rear principal point. This principle is used for finding the principal point with a nodal slide1.

The stationary point depends not only on the optical system, but also on the object and image positions.

A real biconvex “thick lens” operating at 1:1 conjugates has its stationary point in the middle. (CP = -0.5 PP’ and d’ = 2f .)

1. V. J. Doherty, P. D. Chapnik, “Precision evaluation of lens systems using a nodal slide/MTF optical bench,” in

Advanced Optical Manufacturing and Testing II, Proc. SPIE 1591, pp. 103-118 (1991).

'

( ') '

c

c

y PP

CP

f

d y d

(d’)

J. H. Burge

Afocal systems

• For system with object or image at infinity, effect of element motion is tilt in the light.

• Simply use the relationship from the optical invariant:

Where

is the change in angle of the light in collimated space

D0 is the diameter of the collimated beam

0 0n

DF

0 0 i iD D