The Varifocal Membrane Mirror
L. Van Warren
(c) April 4, 1998

Dreaming
Last weekend my wife and I took the kids to the playground. There was a metal picnic bench with thousands of corrugated holes on its surface. This array of holes reminded me of an array of telescopes. I was thinking how nice it would be if we filled the adjacent field with an array of optical telescopes, focused them at the same hunk of sky, and mosaicked all the resulting images together.  Compositing multiple images of the same subject increases the signal to noise ratio [1].

Human Intervention
I was thinking how it would be difficult to assure that the images were coregistered, that is the same size and focal length.  I was thinking about how keeping a human in the loop would make lining up consecutive images simpler [2].

Light Gathering Power
In the midst of all this I remembered an idea I had been thinking about for a good while; A nice way to build cheap telescopes with lots of light gathering power. Light Gathering Power is directly proportional to the area of the mirror, which in turn is proportional to the square of the radius. A large inexpensive mirror would be a great asset for the amateur astronomer since it would enable the viewing of a much fainter set of objects.

Angular Resolution
The cousin of LGP, resolving power is linear proportional to the diameter of the mirror. Resolving power, also known as Angular Resolution determines how much detail can be seen, as in the rings of Saturn for example.

The Big Idea
The idea was to use an aluminized mylar membrane as a mirror, as the active element of a Newtonian reflecting telescope[3]. One could stretch a membrane of this film over a rim, like a drumhead and by withdrawing fluid - air or water - from the backside, produce a mirror of variable focal length.

Reflections On Palomar
Several decades ago, when the 200 inch Palomar mirror was being cast it took 8 months for the glass to cool and over 11 years of shaping in the optical shops of CalTech[4]. At 14.5 tons it was also quite heavy[5]. A membrane mirror can be made in a few hours. A membrane mirror could be quite light. But a membrane might not pull into a perfect parabola. It might not pull into a sphere. I became concerned that geometric aberation would result when the mylar surface was deformed, that uncorrectable artifact would contaminate the resulting images, making them unuseable.

Day Tripping
On the Wednesday following our trip to the park I stopped by a local college. In front of the chalkboard I broached the idea to the chairman of the astronomy department, Andre Rollefson. The chairman shared my concern that geometric aberration was a significant concern. He reminded me that Hubble had been corrected with downstream optics, and jogged my own recollection of those events - I had been at JPL while the modifications to the Wide Field Planetary camera were being made. At this moment, in my mind, the shape of the downstream corrective element became important. Area is plentiful in a membrane mirror, a few points of Fstop lost through corrective elements are easily recovered by making the mirror bigger. The chairman referred me to the assistant planetarium director Charles Hemann. Beginning with my intention of an array of telescopes and ending with suggesting a membrane mirror I repeated the argument, this time on a steno pad. I asked him plainly. "if spherical aberration could be rectified after the fact." He again mentioned that this was, "apparently what had been done on Hubble." I was buoyed up by the realization that I could perhaps build a huge flexible film mirror and then with a downstream optical element correct for the artifact. It was during my sketching at this second meeting that I realized one could digitally trade field of view for signal to noise by simply tiling or compositing the images from an array of optical telescopes.  A mock up of this idea is shown using a conventional image [6]:

Field of View Trade: 4 Mirror Configuration - Wide Angle Tiling of 2 x 2 Array

Signal To Noise Trade: 4 Mirror Configuration - Composite of 2 x 2 Array
I realized that  field of view could be traded for signal-to-noise by using multiple copies of either the same piece of sky, or adjacent pieces of sky.  This provided a digital analog of conventional optical techniques.

Experimental Procedure
Before sinking a large sum of money in an idea, it is often useful to build an inexpensive prototype, just to establish feasibility and to identify trouble spots.
On the following Saturday, I took a coffee can with a hole in the bottom and attached a plumbing fitting. A three foot length of clear 1/4" vinyl tubing tubing connected this fitting to a 50 cc syringe. The can was filled halfway with water.
The first surface was an oven bag which was painted in place by misting with multiple applications of chrome spray paint.

After the painted oven bag proved unsatisfactory (the reflectance was low) aluminized mylar film was stretched over the top and securing with a series of wide rubber bands. This initial method took two people, one to hold the membrane and one to place the rubber band. The mylar was then trimmed, the wrinkles were pulled out and black gaffer's tape was used to secure the film. Black gaffer's tape is a fine fabric tape resembling duct tape, but with a higher quality, less gooey adhesive. It can be torn into strips. When the water was withdrawn using the 50cc syringe, the image appeared bright and reasonably uniform.  This crude varifocal mirror began to work. It was exciting.

A week later, while making the photographs for this article, a much easier method of attaching the membrane to the lip of the can that required no rubber bands, made a much better seal and did not use the gaffer's tape.

Observations
0) Obtaining a watertight seal at both the fitting and the drumhead was nontrivial, requiring several attempts.
1) Stretching the film uniformly over the rim -drumhead style- was quite difficult.
2) Bright chrome spray paint is dull gray when applied to plastic.
3) With the painted oven bag, as the vacuum was drawn, the reflectance appeared to drop as the concavity of the mirror increased.

Further, surface defects from the misting appeared to grow larger as the concavity of the mirror increased. This necessitated a switch to the aluminized mylar.

4) The reflectance of the aluminized mylar also appeared to drop slightly when the film was placed in tension.
5)The smoothness of the mirror, and uniformity of the surface improved with time as the mirror remained in tension.
6) Significant magnfication of the ceiling could be produced by withdrawing water and pulling the membrane down. This was no surprise. It was the intended effect. What was surprising was how quickly and easily the focal length of the mirror could be changed... hydraulically.
7) There were fringes of distortion from small defects in the rim at the edge of the membrane.  These are visible in the title picture above.
8) The unit is sensitive to barometric pressure, air and water temperature.
9) With water in the unit as the principal working fluid, tilted operation is limited.  This restriction can be relieved by using a larger syringe or a simple vacuum pump to withdraw a greater volume of air.

Data Collection
These are crude first-order measurements but one must start somwhere. The first order of business was to measure the shape of the membrane mirror to determine what kind of geometric aberration the stretched film would prefer. A flat metal ruler laid across the top, bisected the tensile mirror.  The image of the ruler yielded a good reflection of the shape of the mirror surface.

To reduce perspective and parallax measurement errors, a piece of Lexan served to provide fixed plane of observation. Using electronic calipers we measured the distance from the ruler to its reflection with the Lexan sheet always enforcing the constraint of constant distance.  The upper edge of the Lexan sheet is barely visible in the left photograph.

Placing the zero point exactly in the center our initial measurement showed that the reflected curve was symmetric to the tolerance of our measurements, about a quarter of a millimeter. With zero representing the center of our mirror, our raw data looks like the following. All units are in mm, read directly off the electronic calipers.

Data Reduction
In this first plot the sense of this mirror is concave downward and we wish to flip the data up so we can have it correspond to the shape of the mirror as we measured it. To flip this graph, we must compute the extremal point in the table and apply a simple linear transform. Searching the data we have the x and y extrema as:
This worked properly. We can now build the new table that flips our data to the correct sense.

This looks more reasonable.

Details
Now we are in a position to compare the measured shape of our mirror to an ideal parabola. First we will define a simple parabolic shape function with a coeffiecient we can tweak. Then we will build a table of values that correspond to the exact x-ordinates we measured above.  Our choice of coordinate system makes the equation for the profile of the idealized shape very simple:

These preliminary results are interesting. The ideal parabola matches our center and edge points exactly to within the tolerances of our measurement. This leaves us only the behavior of the interior points to consider.

Error Analysis
The measured membrane points are above and below the ideal points to the following extents:

The error between the membrane and the ideal parabola is distributed both positively and negatively. Since the greatest error is still within the tolerance of our measurements, to first order,  membrane appears close to the required shape. Typical astronomy quality mirrors are ground to a quarter to half a wavelength of light in accuracy, on the order of 200 nanometers. We are far from that level of accuracy in this, the first exploration of the idea.  The results indicate that further work is justified and our three dollar prototype has served its purpose. More accurate methods of measuring the surface of the membrane will be required in subsequent attempts. But by that time we should be off the coffee can, and onto precision ground, truly circular rings.

Integrating Absolute Error Profile
To determine the best fit parabola we minimize the total error across the face of the mirror subject to the constraint of meeting the center and rim. Repeating the plot of the error above, but this time with absolute value give us:

Computing the area under this curve gives us total error for the mirror in units of mm2.
This corresponds to the area of the error cross section between the ideal mirror and our membrane mirror.
A Brief Excursion Into Modeling
We can go a slight bit farther and simulate the error due to geometric aberration, just to see if it reproduces the observed pattern of behavior[7]. To do this we plot a parabola, and an elliptical section that have the following properties:
1) the shapes agree at the center of the mirror, i.e. y1(0) = y2(0)
2) the shapes agree at the edge of the mirror, i.e. y1(xMax) = y2(xMax)

In the interval of x from -1 to 1 the sections appear to agree fairly closely. Let's look at the error relationship.

The shape of this brief excursion error relationship is not the same as our measured example, so this refined attempt at explaining the error profile is not applicable at this crude level of measurement. Improved Ring Specification

Conclusions
Two models of the membrane mirror were built and demonstrated. Initial measurements were taken and an analysis methodology was developed and presented. Significant refinements in both construction and measuring techniques will have to be applied. This simple proof of concept demonstration implies that the idea is worth pursuing further to the next level of refinement with precision metal rings for membrane suspension. It has been suggested by Charles Rydel of Paris France that concentric high voltage electrodes could be used to improve the shape of the membrane. I propose extending this further by adding a radially symmetric set of point electrodes to give precise control of shape, as illustrated below. The applied voltages could be controlled by a DSP chip and closed loop feedback system to produce a time averaged mirror of useable shape.

Other Questions

For a General Introduction see Ultimate Beginners Guide to Astronomy

References
[1] Yaeger, Larry and Upson, Craig "Combining Physical and Visual Simulation - Creation of the Planet Jupiter for the Film 2010", Computer Graphics, Volume 20, Number 4, 1986, pps 85 - 93. esp p 86 Section 3 Paragraph 2.

[2] Views of the Earth
Summary: The images used were taken by CZCS (Coastal Zone Color Scanner) of NASA Nimbus

7 satellite. The images used were taken by the NOAA weather satellites and knitted together by artist

Tom Van Sant and computer graphicist Lloyd Van Warren for
The Geosphere project

[4] The Palomar Observatory
Summary: Part 1 Part 2 Part 3 Part 4. FORWARD BACK HOME The Pyrex mirror was cast at
the Corning Glass Works in New York in 1934. It took more than 11 years in the optical shops at
Caltech to figure the raw piece of glass into the perfect surface it is today.

[5] Palomar Observatory: Visitor Brochure
Summary: Even more significant is the development of new electronic devices that sense faint signals
of light from distant celestial objects. One of the reasons Palomar Mountain was selected as the site
for the 200-inch telescope was its dark skies that would allow observation of the faintest galaxies
without the interference of city lights.

[6] Astronomy (Science)
Summary: Ben's Space Pages -- Linking Hyper Space to Outer Space. Space
Telescope Electronic Information System (Gopher).

[7] MATH 1509AC --- PROJECT 1
Summary: Namely, all light rays parallel to the symmetry axis are reflected to pass a single focal
point. (i) All light rays parallel to the symmetry axis of a parabolic mirror are reflected through a single
focus; (i) A spherical cap does not bring light rays parallel to the symmetry axis to a single focus.

[8] Aluminised Mylar as a Flux Collector
Summary: Aluminised Mylar as a Flux CollectorThe telescope was first shown at an August 5, 1978, exhibition meeting of the Southeast Group of Astronomical Societies, and again on August 26-27 at the Epsom Borough Show, in the booth of the Ewell Astronomical Society (Maurice Gavin - Worcester Park - Surrey - England vice-chairman).