# Conoscope Lenses

A conoscope is an instrument that measures the angular distribution of light.  This is different from normal lenses (like camera lenses) that measure the spatial (position) distribution of light.  If you were to take a picture of a painting with a normal lens, the picture would closely resemble the painting.  But if you take a picture of a painting with a conoscope, you will get an image that looks similar to one taken with a fisheye lens.  What is special about a conoscope lens is that it precisely maps incoming angles to positions on the image.  This means that if two objects are separated by 5°, they will be mapped to, say, 50 pixels apart, regardless of where they are in the lens’ field of view.  This creates an image that can be very different from a normal lens that maps a distance on the painting to a certain number of pixels on the image.

Why would anyone need a conoscope?  Actually, most people don’t, but some people have light sources and need to measure the angles at which the light comes out of the source.  Others need to measure the angles at which light scatters from a sample when it is illuminated by a laser, that is, they use the conoscopic lens in a scatterometer.  Our first experience with a conoscope was for measuring the light output from an LCD.  The instrument we used was made by Autronics-Melchers, which is now defunct.  Conoscopes for display measurement are still available from Eldim, although they are very expensive.  We can’t match their 88° conoscope (yet), but we can offer much more cost-effective solutions for up to 80°.

## How Conoscope Lenses Work

You want to know how conoscope lenses work?  The best place to start is, of course a picture, so here is a schematic of a conoscope lens:

A conoscope lens consists of the Front End Lenses and the Back End Lenses. The Front End Lenses take all of the rays proceeding from the source at any given angle and focus them to a point on the intermediate image plane. The Back End Lenses simply make a smaller copy of the intermediate image in the appropriate size and location for the image sensor to capture.  If you want more detail, keep reading.

Front End Lenses

A group of rays proceeding at a given angle can also be called a collimated beam. When discussed in these terms, it becomes clear that the Front End Lenses can be thought of as a Fourier Transform lens and the intermediate image plane as the Fourier plane. If this terminology isn’t helpful, you can just ignore it. Some of our customers think in these terms, so we mention it to facilitate communication.  Other customers recognize the Front End Lenses (or the entire conoscope lens) as an F-Theta lens.

A major challenge in the design of the Front End Lenses is that they must perform a precise angle to position mapping. More specifically, the distance from the optical axis to the point where rays entering the lens at a given angle focus must increase linearly with that angle. For example, if the rays entering the lens at 10° come to a focus 10 mm from the optical axis, then rays entering at 20° must focus 20 mm from the optical axis.

Another important thing to note is that the central ray of the beam of light at each angle is parallel to the optical axis in the vicinity of the intermediate image plane. The central ray of a beam is known as the chief ray. When the chief rays from all of the angles are parallel, we have a condition known as telecentricity; the lens is said to be telecentric. Although telecentricity is not required in a conoscope lens, it is very useful when turning the lens into a scatterometer.  For more on that, see our How Conoscopic Scatterometers Work page.

Back End Lenses

The Back End Lenses of a conoscope lens simply reimage the intermediate image plane onto the CCD or CMOS sensor of the camera. The challenge in designing these lenses is that reducing the size to the image to match the size of the sensor decreases the F/# by the same factor. For example, if the intermediate image is 50 mm diameter and the image sensor is 5 mm diameter, we have a magnification of 0.1X. If the beams at the intermediate image plane are F/10, this means that the beams at the camera are F/10 * 0.1 = F/1! If you have ever checked prices for F/1 lenses for your camera, you know that they are expensive. The lens design job becomes easier, and the lens less costly, if the image sensor is larger, but prices for image sensors go up very quickly with size, so the designer must carefully trade off imager size for F/#. This assumes that the intermediate image size is fixed.

Why is the intermediate image size fixed? Why not just make the Front End Lenses smaller? Good questions! If we simply shrink the Front End lenses, the intermediate image also shrinks. But what about the size of the sample? If it shrinks too, everything is OK.  However, for most of our conoscope lenses, this pupil is only 1-2 mm in diameter, enabling the lenses to operate at roughly F/12. Scaling the lenses down by a factor of 2 would bring the sample size down to 0.5-1 mm. That’s getting pretty small! Then why not leave the sample size the same?   The sample is at the pupil of the Front End Lenses, so the size of the sample is the size of the pupil.  The Front End Lenses form a very wide angle lens, they must operate at a fairly slow F/# to produce a reasonably sharp image. With the original 1-2 mm sample size, and the original size lenses, the Front End Lenses operate at roughly F/12.  Halving the size of the lenses without changing the size of the sample would decrease the F/# to F/6.  This would lead to blurrier images on the intermediate image plane (hence lower angular resolution), not to mention decreasing the F/# of the Back End Lenses.  So the short answer is that we can’t decrease the lens sizes because most customers don’t want to decrease the sample size.

Aperture Stop

One final design detail regarding conoscope lenses is that aperture stop of the Back End Lenses is an image of the sample.  In the picture above, the image is not at all sharp, so this is not obvious.  In a conoscope where stray light rejection is important, or where the customer wants to be able to adjust the sample size by placing an iris at the aperture stop, this image must be sharp.  In other cases, it is most cost effective to just let the image stay blurry.  We have designed conoscopic lenses both ways.

## Do you need a Conoscope Lens?

Would a conoscopic lens enable you to make some important measurements?  Take a look at our Conoscope product page and then call us at (651)315-8249 for a consultation to discuss your specific requirements or use our Contact Us page to tell us what you need.