College of Optical Sciences 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 Teaching Other Optical systems engineering and development Fabrication and testing Optomechanics Astronomical Optics Teaching Applied optics classes (Optics laboratory, optomechanics) Other 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 a : tilt Central ray deviation Dy : lateral shift Dq : change in angle

Lens motion tilt decenter (Very small effect)

Effect for lens tilt Can use full principal plane relationships Lens tilt often causes more aberrations than image motion

Mirror motion like lens Dqa = 2a like flat mirror

Motion for a plane parallel plate No change in angle

Motion of an optical system Use principal plane representation Dq s System axis P’ P a Dy PP’ (f = effective focal length) (PP’ = distance between principal points) Pure translation Pure rotation about front principal point If you just tilt your head: Same as single lens a P’ P Dy PP’

Rotation of an optical system about some general point Combine rotation and translation to give effect of rotating about arbitrary point C e(d’) Lateral shift s = CP * ac

Stationary point for rotation Solve for “stationary point”. Rotation about this point does not cause image motion at distance d’. a a P P ’ ’ c c P P C C d 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 Optical invariant: yi qi Optical invariant: This invariance is maintained for any two independent rays in the optical system

Use of invariant for image motion At image plane At element i

The easy part Element i moves, it will cause Dqi = change in angle of central ray (lateral shift Dy is usually small) It is easy to calculate Dqi 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 Image motion from change in ray angle This relationship is easy to remember Dq e 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 Magnification of Image / lens motion NA and Fn based on system focus e Di is “Beam footprint” on element i Di Di

Effect of lateral translation (tilt of PPP) From analysis above: NA and Fn based on system focus ui = NAi Dyi (-)e Magnification for re-imaging :

Example for mirror tilt Tilt a causes angular change Which causes image motion “Lever arm” of 2 Fn Di ( obvious for case where mirror is the last element) Dq e a 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: Where Dq0 is the change in angle of the light in collimated space D0 is the diameter of the collimated beam

Thank you! 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 This gives an easy way to relate surface figure to image blur. More on this later… Thank you!