Go up to 6 Getting good extra-focal images.
Go forward to 6.2 Special considerations for astronomical telescopes.
6.1 General considerations
The program ef , is designed to compute optical
wavefronts from extra-focal images. The algorithms used in the
program are based upon the fact that an aberration in the
optical wavefront will change the shape and intensity
distribution within a de-focused pupil image.
More explicitly we use the optical transport equation:
Where
I(x,y,z)
is the irradiance distribution along the beam and
W(x,y,z)
is the Wavefront surface referenced to the z (propagation) direction.
The quality of the results obtained is dependent on both the
characteristics of your optical system, and the care with which you
take measurements.
The basic measurements needed by the program are the intensity
distribution in two extra-focal planes.
In
the above figure
We show diagrammatically the placement of extra-focal planes
around a focal plane. Note that the distance to each
extra-focal plane (from the image) plane is the same, this
is not essential, but the closer you come to this ideal situation,
the better your wavefront measurements are likely to be.
There are many possible ways to obtain these extra-focal images,
the most straightforward being simply to translate your camera
along the optical axis of your imaging system.
With some optical systems, you may be able to adjust the focus of the
optics.
However re-focusing may induce different aberrations from your optical
system at each extra-focal plane and thus adversely affect your wavefront
measurement.
A common case of this, is re-focusing a cassegrain telescope, which
introduces variable spherical aberration, and possibly coma if the
focus mechanism is not accurately aligned.
In the following sections we will discuss the issues
associated with taking extra-focal images. We have
split the discussion into several sections, but none
of the considerations is completely orthogonal to the
others.
It is often beneficial to reduce data taken under
several different conditions to quantify the
wavefront measurement error.
Extra-focal images should be placed symmetrically either side of the
stop in your optical system. The program makes that assumption
that the pupil stop is close to infinite conjugate to the image
plane. In the current version of the program it is assumed that
the optical stop is circularly symmetric. If you wish to
obtain wavefronts from systems lacking circular symmetry please
discuss your needs with Laplacian Optics Inc. .
Some optical systems have multiple planes which nearly define the
pupil stop. This can lead to non circular symmetry if the optical
system is mis-aligned. A good example is an infrared astronomical
telescope, where the secondary mirror normally defines the optical stop.
However the primary mirror edge is only slightly beyond the secondary
edge. Under these circumstances a telescope mis-alignment can result
in the optical stop being defined by the secondary mirror in some places
and the primary mirror in others. The mis-alignment will cause the
effective optical stop to be non circularly symmetric. The retrieved
wavefront will therefore be in error, the main effect being to
reduce the accuracy of low order terms such as astigmatism and coma.
The best course of action in this case would be to use the
measured wavefronts to calculate the adjustments needed to align
the optical system. If the mis-alignment is large, several iterations
might be required in order to achieve adequate alignment,
Depending upon your optical system, the physical size of
the extra-focal image may be related to the extra-focal
distance.
We generally recommend using a lens system to take extra-focal
images, partly because such a system allows one to decouple
the pupil image size from the extra-focal distance, please
contact Laplacian Optics Inc. . for more information.
The size of the reconstructed wavefront is not directly related extra-focal
image diameter, but is set independently as a part of the data reduction
process. In general the resolution on the reconstructed wavefront is
substantially lower than that of the extra-focal images.
There are two basic factors which must be considered when attempting to
set the size of the extra-focal images:
Due to the transformation of the extra-focal images that takes
place whilst this program is running, it is wise to have
several times more pixels across the extra-focal image, than is
required in the wavefront.
One should on no account try to reconstruct a wavefront of
higher resolution than the extra-focal image size.
We recommend about 100 to 200 pixels across the diameter of an
extra-focal image.
The program will work with smaller images, but
will tend to give less accurate results.
Images larger than the recommended size will work well, but may
slow down the data-reduction process.
As with all imaging applications one would like to ensure that
the extra-focal images are properly sampled. However depending
upon the optical configuration this may not be feasible.
Obviously any under sampling of the extra-focal images
will potentially introduce aliasing errors which will
reduce the accuracy of the wavefront reconstruction.
Most optical surfaces contain little high frequency aberration so
the loss of accuracy due to aliasing is not usually significant.
Hence with most optical systems and accuracy requirements,
under sampling is likely to produce acceptable results.
As a general rule it is recommended that you record images
at several different extra-focal distances and/or image samplings.
Any large differences in the wavefront derived from images with
different sampling or extra-focal distances indicates a problem.
The program ef core algorithm is based on the
fact that wavefront deformations produce brightness variations
within the extra-focal pupil image which are proportional to
the Laplacian of the wavefront. This relationship is only
approximately linear, and most of the work performed by
ef is to apply corrections for non-linearity.
The more severe the non-linearity the more error is generated
in the reconstructed wavefront.
The main trade-offs to be considered when choosing an extra-focal
distance is the following:
- Closer to the focal plane
Non-linearity grows, wavefront signal grows (SNR better),
wavefront spatial resolution lower.
- Farther from the focal plane
Non-linearity reduced, wavefront signal reduced (SNR worse),
wavefront spatial resolution improves.
In practice the trade-off between SNR and linearity is usually
the most important.
The size of the signal scales as:
sig = w*f*f/l
Where w is the magnitude of the wavefront error, f is the f-ratio
of the optical system you are measuring, and l is the
extra-focal distance you are using.
A good heuristic to employ is that if the curvature signal
(intensity difference inside extra-focal image) significantly exceeds 25%,
you need to increase the extra-focal distance to avoid
excessive non-linearity. The program will still work
with larger signal strengths, but will give reduced accuracy.
Note the signal may be observed by loading two extra-focal
image and hitting the reset followed by the iterate buttons
in the Reduce :: Run reduction manually menu sub-window.
After the iteration has finished the signal will be displayed in
the left image window.
We recommend that data for this program always be taken
with a slow scan CCD with at least 10 bits of dynamic range.
With even an inexpensive (less than US$1000) amateur grade
slow scan CCD the image SNR will be dominated by photon noise
when the pixel intensity reaches more than 10% of full well.
We recommend exposing extra-focal images to give a
peak intensity of between 10% and 50% of full well, and using
dark/background and flat field images if available.
In our experience most commercial grade frame grabber/video
camera combinations give only 5 to 6 bits of dynamic range
which is barely adequate for taking reasonable extra-focal
images. Worse still, most TV cameras have a nonlinear transfer
function or gamma correction, which seriously compromises the
accuracy of any extra-focal images that you record.
Lastly the automatic gain control found on
some video cameras can change the image gain
within a single image thus distorting the image brightness
distribution, particularly when imaging a very contrasty
object, as the extra-focal image should be.
If you do use a frame grabber to acquire images, you should
make the pupil images as large as possible to average the
signal over many pixels, turn off automatic gain control,
and average several image frames in software (or using a hardware
16 bit accumulator) to improve SNR.
Under most situations it should be possible to build wavefront
sensor optics which do not introduce significant aberrations
into the measures wavefront. This is possible, because the
image quality required in the extra-focal image is quite
modest. The most difficult problem with most measuring
systems is to control the effects of seeing within the
system being measured. The curvature sensing approach as
implemented by the ef program allows two approaches
to controlling this source of error.
As with traditional measurement systems, one can take short
exposure images to freeze any turbulence aberrations, then
average several measurements to reduce the effect of these
time varying aberrations. To
do this with the curvature system requires setting up the
wavefront sensor so that it can record both extra-focal images
simultaneously.
The preferred approach to eliminating turbulent seeing effects
is to integrate each extra-focal image long enough to average
out the effects of seeing. The amount of time required to to
this depends on the seeing, but is typically of the order
of 1 minute or longer. For the special case of atmospheric
errors see
later.
A neutral density or narrow band filter can be used directly in front of
the camera to reduce problems with background light.
Small variations in filter should have little impact on the wavefront
measurement, provided both extra-focal images fall on the same set of
camera pixels (pass through the same piece of filter), and a lens system
is used so that images need not be inverted.
When seeing is a consideration we recommend taking several sets of
data so that you can compute directly the random errors on the measured
wavefront.
One of the main advantages of the curvature wavefront technique, is its
simplicity and low cost of implementation. A related drawback to the method
is that it can be adversely affected by various pathological optical problems.
Such problems can usually be mitigated by either adding additional optics
to the test system, or by suitable configuration of the software.
The main purpose of this section is to make you aware of potential
problems to enable you to best apply this software.
One of the most common optical aberrations encountered in optical
fabrication is errors on the edges of an optical component.
In operation this can lead to light scatter from the edge of the optical
pupil.
Since the program uses the shape of the pupil to derive
boundary conditions for solving the Poisson equation,
light scatter at the pupil circumference can prove problematical.
The most noticeable effect of this problem is difficulty in
accurately finding wavefront Zernike terms with zero
curvature.
Most commonly this produces inaccurate measurements for
the astigmatism terms.
The first thing you should do under these circumstances is to
increase the extra-focal distance. You should try to get
far enough from focus to guarantee that the light from
the rolled edge is not in the caustic Zone. This will
reduce the SNR for all other aberrations, but there is
no other choice. If you cannot increase the extra-focal
distance sufficiently you may try the reduction sequence
rolled available under the
Edit :: Select reduction sequence is tuned to give
better results with a moderately rolled wavefront edge.
Be warned however that this does not always work, and even if it
does, the reconstructed wavefronts will be less reliable.
For severe cases of turned edges, we would recommend excluding the
damaged part of the wavefront by using a physical pupil stop.
This is likely to improve the imaging performance of your optics
as well!
In systems where significant chromatic aberrations may be present,
you should not use wide band imaging for the extra-focal images.
Typically chromatic aberration will produce a signal which looks
very similar to spherical aberration.
Ideal in this case would be to measure the extra-focal images using
several narrow band filters, which would allow determination of
the systems chromatic behavior.
When using a monochromatic source to measure extra-focal images, you may
observe speckle or low-level fringing in the extra-focal image plane, due
to reflections or scattering within the optical system.
If the structures are properly sampled, and fixed with respect to the
extra-focal image coordinates, they will theoretically, have no effect
when measuring small aberrations.
In practice these conditions are unlikely be met, so you should treat
speckle and fringing as a source of measurement noise.
It is probably unwise to under-sample fringing structures, since this
could give large low frequency errors, under-sampling speckle will be
less severe.
A possible optical solution is to introduce time varying random
aberrations (eg seeing) and allow these to average out some of the speckle
(and possibly fringing).
Laplacian Optics Inc. Email: laplace@laplacian.com