Converting Zernike coefficients into useful values

[theot...@gmail.com]

I'm looking for a tutorial (primer) on how to use the Zernike coefficients obtained from a Shack-Hartmann wavefront sensor.
For example, how to I use the x-tilt coefficient to work out how far my beam will shift?

[Phil Hobbs]

On 04/25/2013 04:01 AM, theot...@gmail.com wrote:
> I'm looking for a tutorial (primer) on how to use the Zernike coefficients obtained from a Shack-Hartmann wavefront sensor.
> For example, how to I use the x-tilt coefficient to work out how far my beam will shift?
>

Is this an adaptive optics application?

Cheers

Phil Hobbs

--
Dr Philip C D Hobbs
Principal Consultant
ElectroOptical Innovations LLC
Optics, Electro-optics, Photonics, Analog Electronics

160 North State Road #203
Briarcliff Manor NY 10510

hobbs at electrooptical dot net
http://electrooptical.net

[Salmon Egg]

In article <1fd34472-0c42-4f48...@googlegroups.com>,

While not an expert on the subject, I would guess you do what you would
do to expand any arbitrary function using an approprcate complete set of
orthonormal functions.

To me, the term "wavefront sensor" implies that you have data that tells
you the wave front error as a function ต(x,y) of x and y. The trouble
may be that you only have samples of ต. That means you either use an
integration grid at those sample locations or some interpolation to get
values at points needed for numerical integration.

Look up some combination of Sturm-Louiville (sp), harmonic analysis,
fourier series. IIRC there was a FORTRAN program out of U of Arizona
that used Zygo interferometer data and converted it to aberration
coefficients.

--

Sam

Conservatives are against Darwinism but for natural selection.
Liberals are for Darwinism but totally against any selection.

[theot...@gmail.com]

I guess it is a really simple one, yes.
I have a rectangular transmissive optical element that is heated from one side, setting up a temperature gradient in the x-direction. I have a Shack-Hartmann (SH) set up to measure the phase on this element, and do indeed see the "tilt-x" aberration developing in the x-direction. My SH gives me the amount of tilt as a Zernike coefficient. My problem is to figure out how much (in milliradians, for example) a beam passing through the element will be deflected.
That is the immediate need. I would also be interested in a more thorough discussion on the effects of a measured wavefront on a transmitted beam in a textbook or tutorial-type article.



On Thursday, April 25, 2013 4:45:26 PM UTC+2, Phil Hobbs wrote:

[theot...@gmail.com]

Thanks for the interest. I hope my reply above gives a little more information.



On Friday, April 26, 2013 4:36:21 AM UTC+2, Salmon Egg wrote:
> In article <1fd34472-0c42-4f48...@googlegroups.com>,

[Salmon Egg]

In article <f99cc981-cc22-473e...@googlegroups.com>,

theot...@gmail.com wrote:

> Thanks for the interest. I hope my reply above gives a little more
> information.
>
>
>
> On Friday, April 26, 2013 4:36:21 AM UTC+2, Salmon Egg wrote:
> > In article <1fd34472-0c42-4f48...@googlegroups.com>,
> >
> > theot..me@gmail.com wrote:
> >
> >
> >
> > > I'm looking for a tutorial (primer) on how to use the Zernike
> > > coefficients
> >
> > > obtained from a Shack-Hartmann wavefront sensor.
> >
> > > For example, how to I use the x-tilt coefficient to work out how far my
> > > beam
> >
> > > will shift?
> >
> >
> >
> > While not an expert on the subject, I would guess you do what you would
> >
> > do to expand any arbitrary function using an approprcate complete set of
> >
> > orthonormal functions.
> >
> >
> >
> > To me, the term "wavefront sensor" implies that you have data that tells
> >

> > you the wave front error as a function µ(x,y) of x and y. The trouble
> >
> > may be that you only have samples of µ. That means you either use an

> >
> > integration grid at those sample locations or some interpolation to get
> >
> > values at points needed for numerical integration.
> >
> >
> >
> > Look up some combination of Sturm-Louiville (sp), harmonic analysis,
> >
> > fourier series. IIRC there was a FORTRAN program out of U of Arizona
> >
> > that used Zygo interferometer data and converted it to aberration
> >
> > coefficients.
> >
> >
> >
> > --
> >
> >
> >
> > Sam

I expect that you will have some problems getting Zernike coefficients
over a rectangular domain. The theory I know of, and I may be out of
date on the topic, is that Zernike functions are based upon using a
circular aperture. I would expect that it would be necessary to limit
the boundary to a circle. This prevents various artifacts from
truncation. Ptherwise Gibbs phenomenon will show up. That is, you will
probably have to use a circular domain of integration.

Another thing I learned from this post having nothing to do with the
topic is the way encoding of the message affects it Appearance.

My Macintosh system allows me to easily generate some Greek symbols in
line with other text. In this case, the Greek mu did not transmit in a
universal form. It is read by my Mac with no problem. After traveling
through other computers on repost, I did receive a mu. As a consequence,
however, it appears that at least one system changed the coding/ The
result was that one form of Unicode kept the mu but also greatly
diminished the font size. When changing back to a Latin decoding, the
font regained its size but the mu was not interpreted.

I apologize vof this, I have complained when others had done the same
thing.

[Phil Hobbs]

On 04/26/2013 02:40 AM, theot...@gmail.com wrote:
> On Thursday, April 25, 2013 4:45:26 PM UTC+2, Phil Hobbs wrote:
>> On 04/25/2013 04:01 AM, theot..rme@gmail.com wrote:
>>
>>> I'm looking for a tutorial (primer) on how to use the Zernike
>>> coefficients obtained from a Shack-Hartmann wavefront sensor.
>>
>>> For example, how to I use the x-tilt coefficient to work out how
>>> far my beam will shift?
>>>
>>
>> Is this an adaptive optics application?
>>
>> Cheers
>>
>> Phil Hobbs

> I guess it is a really simple one, yes. I have a rectangular
> transmissive optical element that is heated from one side, setting up
> a temperature gradient in the x-direction. I have a Shack-Hartmann
> (SH) set up to measure the phase on this element, and do indeed see
> the "tilt-x" aberration developing in the x-direction. My SH gives me
> the amount of tilt as a Zernike coefficient. My problem is to figure
> out how much (in milliradians, for example) a beam passing through
> the element will be deflected. That is the immediate need. I would
> also be interested in a more thorough discussion on the effects of a
> measured wavefront on a transmitted beam in a textbook or
> tutorial-type article.
>

I'm not the biggest fan of either Zernikes or Shack-Hartmanns, although
for tilt a S-H will be fine.

Zernikes are only orthogonal on a circular disk, so any vignetting or
odd-shaped pupils make them non-orthognal and often ill-conditioned, so
a slight shadow will make your computed coefficients go all over the
place. Seidel coefficients have much less of a problem with this,
although they're ill-conditioned at high orders as well.

Shack-Hartmanns are fast, low-resolution sensors of local wavefront
tilt, from which they have to reconstruct the actual wavefront by
integrating, which is is even less accurate. Their speed makes them the
bee's knees for adaptive optics, but I wouldn't use one for testing an
instrument unless I was desperate.

The Zernike polynomials are defined e.g. at
http://mathworld.wolfram.com/ZernikePolynomial.html , which quotes Born
and Wolf.

Cheers

Phil Hobbs

PS:

On Usenet it's customary to add one's reply to the bottom of the
(appropriately snipped) context, so that it's easier for others to read
sequentially. (Email uses the opposite convention, because only those
participating in the discussion are reading it.)

[anorton]

<theot...@gmail.com> wrote in message
news:889e9f0e-55a3-4bbf...@googlegroups.com...

>I guess it is a really simple one, yes.
>I have a rectangular transmissive optical element that is heated from one
>side, setting up a temperature gradient in the x-direction. >I have a
>Shack-Hartmann (SH) set up to measure the phase on this element, and do
>indeed see the "tilt-x" aberration developing >in the x-direction. My SH
>gives me the amount of tilt as a Zernike coefficient. My problem is to
>figure out how much (in >milliradians, for example) a beam passing through
>the element will be deflected.
>That is the immediate need. I would also be interested in a more thorough
>discussion on the effects of a measured wavefront on a >transmitted beam in
>a textbook or tutorial-type article.

As others have said, fitting Zernikes to a square aperture can be
problematical, it depends on how good the algorithms are handling this.

There are a couple of different versions of the zernike polynomials, and you
have to know which the sotware is using. sometimes Z1 polynomial is defined
as 2rho*cos(theta) and sometimes just rho*cos(theta). rho is the
normalized semidiameter of the aperture, so if it is using the later
definition, and the Z1 coefficient equals 1, the wavefront tilt is just 1
wave per semidiameter. With a shack hartmann sensor you also have to
confirm they are not expressing wavefront height in terms of microns which
is the more natural number for it to calculate.

Actually the SH sensor is really measuring wavefront slope at many points
across the aperture, and if slope is what you want perhaps there is a way to
acess that data directly. The wavefront height and coefficients have to be
calculated from the slope.

Often when measuring wavefront error, the tilt and focus are subtracted
because these usually depend on the precise alignment of the optic in the
test set-up. So, when you heat your element, how do you know the changes you
are seeing are due to changes in the element itself or due to changes in the
mount holding the element? Do you care?
--
Adam Norton
Norton Engineered Optics
www.nortonoptics.com

(Remove antispam feature before replying)

[Skywise]

Salmon Egg <Salm...@sbcglobal.net> wrote in
news:SalmonEgg-ACEAF...@news80.forteinc.com:

> Another thing I learned from this post having nothing to do with the
> topic is the way encoding of the message affects it Appearance.
>
> My Macintosh system allows me to easily generate some Greek symbols in
> line with other text.

I think it's really up to the particular newsreader software
as I had no problem with the mu. I'm using Xnews on a windows
system.

Then there's email. If only we could get people to STOP using
html... ughhh. I don't care about fancy wallpaper backgrounds.
Email is email. A webpage is a webpage. One is not the other.

:)

Brian
--
http://www.skywise711.com - Lasers, Seismology, Astronomy, Skepticism
Sed quis custodiet ipsos Custodes?

[Salmon Egg]

In article <udKdnVWbccOvTOfM...@earthlink.com>,

It really boils down to what the purpose for using the Zernike expansion
is. It makes sense for circular apertures because the lower orders
correspond to various named aberrations such as coma and astigmatism. It
might make sense to expand wavefront errors over a RECTANGLE. Thus, one
might still apply names such as tilt to terms piston, tilt, cylinder,
sphere, etc. But these will not be Zernike coefficients

Rectangular harmonics based on product of sines and cosines might work.
For instance, multimode metallic microwave waveguides can be analyzed
that way. Legendre polynomials could be good for expansions over a range
from -1 to +1. As functions of x or y they could be made suitable for a
rectangular expansion even though they are usually used with as
components of spherical harmonics. That certainly should lead directly
to piston and tilt.

I do not know what is available in the literature to help carry such a
procedure out. Nevertheless, my guess it is just a matter of classical
applied mathematics.

[Joe Gwinn]

In article <SalmonEgg-EAA01...@news80.forteinc.com>,

Are we overthinking this? If all that's needed is to detect tilt, and
the OP doesn't care about the various named aberrations, just perform a
least-squares fit to a plane of constant phase from the Shack Hartmann
sensor.

Joe Gwinn

[Salmon Egg]

In article <260420132127587869%joeg...@comcast.net>,

Joe Gwinn <joeg...@comcast.net> wrote:

> Are we overthinking this? If all that's needed is to detect tilt, and
> the OP doesn't care about the various named aberrations, just perform a
> least-squares fit to a plane of constant phase from the Shack Hartmann
> sensor.

That make great sense. The OP has not told us, or I have not understood,
exactly what he was looking for.

BTW, the program I was trying to recall for analyzing fringe data as
obtained by Zygo interferometers was called, fittingly, FRINGE from the
U of Arizona. That dates me, because I am pretty sure modern
interferometers can produce wavefront error maps without going through
an intermediate step of obtaining fringes. If the U of A group is still
active, they may have just the kind of software the OP needs.