Transcript Slide 1

An easy way to relate optical element
motion to
system pointing stability
Jim Burge
College of Optical Sciences
Steward Observatory
University of Arizona
Prof. Jim Burge
• Room 733 (in the new building)
• Research
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Optical systems engineering and development
Fabrication and testing
Optomechanics
Astronomical Optics
• Teaching
– Applied optics classes (Optics laboratory, optomechanics)
• Other
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Sailing, diving, fishing in San Carlos, Mexico
Mountain biking
Ultimate frisbee
Beer brewing
Goals for this talk
Provide
• Basic understanding of some optical/mechanical
relationships
• Definition, application of the optical invariant
• Useful, easy to remember equations to help make your
life easier
Motion of optical elements
• 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)
Lens decenter
• All image points move together
• Image motion is magnified
What happens when an optical element
is moved?
To see image motion,
follow the central ray
Generally, it changes in position and angle

Element motion
s : decenter
 : tilt
Central ray deviation
y : lateral shift
 : change in angle
ay
r
d
e
at
devi

y
Initial on-axis ray
s
Lens motion
decenter
tilt

s
 

f

s
 s 
f
s

f
 
(Very small effect)
Effect for lens tilt
• Can use full principal plane relationships
• Lens tilt often causes more aberrations than image motion
Mirror motion

s

s
f
like lens
s
   2
f
 = 2
like flat mirror
Motion for a plane parallel plate

y
Plane parallel plate
thickness t
index n
 t  n  1
y 
n
No change in angle
Motion of an optical system
Use principal plane representation
Pure rotation about
front principal point
Pure translation
System axis
P
P’

y
s

s
f
y  0
PP’
 
(f = effective focal length)
Same as single lens
y    PP '
  0
(PP’ = distance between
principal points)
If you just tilt your head:
Rotation of an optical system
about some general point
•
Combine rotation and translation
to give effect of rotating about
arbitrary point C
c
C
P
P’

y
d’
Lateral shift s = CP * c
s
 
f
y   c  PP '
CP
   c
f
e(d’)
Stationary point for rotation
• Solve for “stationary point”. Rotation about this point does not
cause image motion at distance d’.
c
e (d ')  y  d ' 
0
C
d’
CPstationary
f
   PP '
d'
Thin lens (PP’=0)
stationary point at P = P’
Object at ∞ (f = d’)
stationary point at P’
Otherwise it depends on separation of principal planes and image conjugates
Optical Invariant
Marginal ray
Marginal ray –
on axis ray that goes
i
yi
through edge of plane
of interest
yi y i
Chief ray –
off axis ray that goes
through center of the
plane of interest
(typically the stop.)
Optical invariant:
Chief ray
Plane of interest
Position i
I  ui yi  ui yi
This invariance is maintained for any two independent rays in
the optical system
i i
Di
Use of invariant for image motion
Element i
NA and Fn based on
system focus
Light from
point on axis,
Bundle defined
by aperture
Off-axis light
is ignored
Element i
“Beam footprint”
on element i
Nominal marginal
rays at element i
ui = NAi
NA and Fn based on
system focus
Di
Image
shift
e
Perturbed central ray
from element i
yi
Image
shift
e
uAt
i
i  image
I  NA  e 
yi  yi
Nominal marginal
rays at element i
ui = NAi
Perturbed central ray
from element i
At element i
“Beam footprint”
on element i
Di
I
i  NAi yi
2
yi
ui  i
yi  yi
plane
e
2 Fn
NAi
e  Fn Di i 
yi
NA
The easy part
NAi
e  Fn Di i 
yi
NA
Element i moves, it will cause i = change in angle of
central ray
(lateral shift y is usually small)
It is easy to calculate i
Image motion is proportional to this
All you need is
Fn final focal ratio
Di beam footprint for on-axis bundle
Example for change in angle
e  Fn Di i
Image motion from change in ray angle
This relationship is easy to remember
D

e
e  f 
 Fn D
f = FnD
Reduces to a simple example for a single lens!
Effect of lens decenter
Decenter s causes angular change
Which causes image motion
 i 
e  Fn Di
Magnification of Image / lens motion
s
fi
s
fi
e  Fn
s
fi
NA and Fn based on system focus
e
Di is “Beam footprint”
on element i
Di
Di
Di
Effect of lateral translation
From analysis above:
(tilt of PPP)
NAi
e 
yi
NA
NA and Fn based on system focus
e 
NAi
yi
NA
yi
ui = NAi
e
Magnification for re-imaging :
e
NAi

yi
NA
Example for mirror tilt
Tilt  causes angular change
i  2
Which causes image motion
e  2Fn Di i
“Lever arm” of 2 Fn Di
( obvious for case where mirror is the last element)
e

d
e  d 
 Fn D
 2 Fn D

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 invariant:
e
Fn
 D0 0
Where
 is the change in angle of the light in collimated space
D0 is the diameter of the collimated beam
D00  Di i  2NAi yi
Other useful things
• Useful for pupil image as well. Just be careful to use correct
definitions
• Also use this to relate slope variation across pupil to the size of
the image blur
e (x ,  y )  Fn Di i (x ,  y )
• This gives an easy way to relate surface figure to image blur.
More on this later…
Thank you!