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. So what is a conoscope used for?

In the past, conoscopes were developed to evaluate liquid crystals. These LCDs are characterized by variable brightness depending on angle of view and exhibit variations in contrast and color-shift. All these qualities can be mapped by attaching a conoscope to a camera to produce an image displaying a circle that represents a near-hemisphere of performance, with each pixel representing a combination of polar angle and azimuth. 

Today, a conoscope lens is useful because 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 (this particular type of testing has exciting implications for new tech, which you can read about on our Light Source Measurement page). 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°.

Variable Conjugate

Some conoscopes are designed to work at multiple object distances, which is often the case with VR/AR inspection. A conoscope capable of this is called a variable conjugate lens, or variable object distance lens. To inspect VR/AR the conoscope must be able to accommodate both near-sghted and far-sighted users. The conoscope must be able to “focus” to adjust for varius object distances. Most systems for this application cover object distances from 250mm in from of the lens to infinity. But some are capable of focusing from -100mm (I.e. inside the lens), to infinity, and to +100mm (I.e. in from of the lens). Optometrists speak in terms of Diopters, which have units of inverse meters. Zero Diopters (0D) corresponds to a plane parallel plate or an inifinite object distance, and 10 Diopters (10D) corresponds to 0.1 meter, or 100mm. In these units, a normal system adjusts from 0D to 4D, whereas a wide range system ranges from -10D to +10D.

Specifying a Conoscope

When designing a conoscope, it is important to know some of the basic parameters that the lens will require to measure the light source accurately. We have designed the calculator below so you can determine these basic parameters from a few simple specifications. If you know your required image height, maximum desired acceptance angle, working distance and the diameter of the device under test, this calculator can help you determine the focal length, f/# and diameters of your conoscope lens.

Essential Design Considerations

It’s also vital to be aware of several essential design considerations. When specifying a conoscope, the most important attribute to remember is the Lagrange invariant, or conservation of etendue. In plain terms, violating these rules would be similar to fitting 10 pounds of material into a 1-pound bag. Basically, you cannot have a wide angle conoscope with a large pupil connected to a small image sensor. 

As applied to conoscopes, the equation is: 

Φp * sinΘ = Φi * NAi, 

In this equation, Φp represents the diameter of the entrance pupil, Θ is the half-field of view of the conoscopeΦi is the diameter of the image and NAi is the numerical aperture of the lens. The NA of a lens is approximately half the inverse of the F/# (= 0.5 / F/#). 

Understanding this is critical when specifying a conoscope. Once the aperture (or sample size) and field of view are chosen, then either the image size or the F/# of the lens can be chosen, but not both. Take, for example, a conoscope with an 80° half-field of view and a 4-mm diameter aperture. Designing the lens for a 1/3.2-inch (3 x 4 mm) image sensor will require a lens with an NA of 1.3. Because lens designs start to become challenging at NA 0.25 (F/2), meeting this hypothetical requirement would be an impossibility. 

The working distance (WD) from the sample or aperture to the front of the conoscope lens is another specification that can raise potential issues. Engineers who design inspection systems like to have a lot of room between the device under test and the lens. That makes sense. Unfortunately, when they request a working distance of 50 mm and a half-field of view of 75°, a quick calculation shows that the first surface of the lens must be about 380 mm in diameter, as demonstrated by the following equation: 

Φ1 = Φp + (2 * 50 * tanΘ) = 6 + (2 * 50 * tan(75°)) = 385 

Even worse, the first surface cannot bend the rays very far, which requires the conoscope to be about twice the diameter of the first surface. Faced with the prospect of a multimillion-dollar lens, most engineers would instead favor a more reasonable working distance. 

While these first two rules of design are based on basic physics and geometry, a third rule is more of a heuristic and provides a rough idea of the challenges of designing a conoscope. The third rule is defined by the following equation. 

x = Φp * Θ / OD 

where OD is the outside diameter of the conoscope and x represents the degree of difficulty. If Θ is given in degrees in this equation, then a value of less than 2 for x represents an easy design while values exceeding 5 or more correspond to a difficult one. For example, a conoscope with a half angle of 60 degrees, an outside diameter of 60 mm, and an entrance pupil diameter of 4mm would have an x factor of 4.  

There are also other factors that can make design more difficult, including a large (>70°) field of view, a wide wavelength range, a large NA (fast F/#) or a large zoom range would add to the difficulty. 

Image Sensor Size Considerations

Choosing the correct image sensor for a conoscope can be a difficult task. It is tempting to design for as cheap of a sensor as possible to keep costs down, but before this decision is made it is important to consider everything that will be affected by the choice. 

The first things to decide are the required angular resolution for the lens, the entrance pupil diameter, and the horizontal field of view. Once these are known the image sensor selection process can begin. For example, say we need a conoscope with a ±60° HFOV, an entrance pupil diameter of 4mm, and a minimum angular frequency of 20 cyc/deg (0.025° resolution). 

Here are two potential image sensors for the system: a small sensor (Sony IMX183) and a large sensor (Gpixel GMAX3265). Below is a table showing the metrics for each sensor: 

Where R is the ratio of image diameter to object angle; νN is the Nyquist frequency in cyc/mm; Px is the pixel size in μm and νθ is the Nyquist frequency in cyc/deg.

From this initial evaluation it would be easy to conclude that the IMX183 is the best option because it meets the required 20 cyc/deg with a cheaper image sensor. However, this conclusion does not take into account all of the factors that are necessary during a conoscopic system design. First, the conoscope lens design for the IMX183 will need to resolve a higher spatial frequency to meet the 20 cyc/deg requirement than the GMAX3265 design will. Second, the F/# of the lens will be much lower for the smaller sensor, which will make the lens design more difficult and therefore more expensive. Below is a comparison of these metrics for the two conoscope designs. 

Where EPD is the entrance pupil diameter; f is the focal length of the lens (for a conoscope y’ = f*θ when corrected for f-theta distortion); θr is the angle in radians; F/# is the f-number of the lens; νθ is the angular frequency specification and v is the spatial frequency.

As seen in the chart, the spatial frequency required to resolve 20 cyc/deg is much lower for the GMAX3265 (80cyc/mm vs 183 cyc/mm) which will make the lens design simpler. The F/# is also higher which will also simplify the lens design. Simpler lenses require fewer glass elements and are easier to assemble, so they cost less. 

Using a larger image sensor will often result in a test system that has superior performance and a lower overall cost than an equivalent system designed for a smaller image sensor.

As always, if you’d like help specifying your conoscope, feel free to contact us and one of our engineers will be glad to assist you. 

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:

Conoscope Lens Schematic

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. 

The first group of lens elements in a conoscope includes an external pupil and maps the angle of incoming light onto a linear distance on the internal image plane. In other words, as the angle between a light ray and the optical axis of the lens increases, so does the distance from the optical axis to the point where the ray intersects the image plane. Doubling the incoming angle also doubles the distance from the axis to the intersection, which is known as the image height. 

Often, the image height for the maximum ray angle is too large to accommodate an affordable image sensor, which requires the size of the image to be reduced. That is the main function of a conoscope’s second lens group. The second group removes residual image errors left by the first group and may also include the aperture stop for the lens if there is no physical aperture in front of the first group.   

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? Go to our Display Measurements page to read more about possible applications or take a look at  our Conoscope 
product page to learn about our stock options. Give us a call at (651)315-8249 for a consultation to discuss your specific requirements or use our contact us page to tell us what you need.