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 a circular image that depends on the texture of the painting and how it is illuminated. It will bear no resemblance to the painting itself.
Our article further expands on these and other key points beginning after the introduction.
Conoscope lenses map incoming light angles to image plane positions, making them essential for measuring angular light distribution in displays, LEDs, DOEs, and scattered light applications.
A typical conoscope includes front-end lenses that perform angle-to-position mapping and back-end lenses that resize the intermediate image for capture on compact sensors.
Key optical design constraints, like the Lagrange invariant, working distance, and sensor size, govern what’s physically possible and cost-effective when specifying a conoscope (see our conoscope lens cost guide for what this means in practice).
Using larger image sensors simplifies lens design and can reduce system cost and complexity, which is why proper sensor selection is just as critical as lens geometry.
A conoscopic lens precisely maps incoming angles to positions on the image. This means that if two objects are separated by 5°, they will be the same distance apart in the image, regardless of where they are in the lens’ field of view. Attaching a camera to a conoscope will produce a circular image that captures light over almost a hemisphere, with each pixel representing a combination of polar angle and azimuth.
Why would anyone need a conoscope? Most people don’t, but if you’re working with light sources, you might. A conoscope is used when you need to measure the amount of light as a function of emission angle. This applies to sources like LCDs, OLEDs, LEDs, laser-excited phosphor (LEP) modules, diffractive optical elements (DOEs), and even surfaces illuminated by collimated beams such as diffusers or other illuminated samples.
For example, DOEs are used to project infrared dot patterns in facial recognition systems and must be measured for pattern and brightness uniformity and angular distribution. LEDs and LEP sources require characterization of their emission profiles for optical system design. If you’re interested, we go into more detail on our Light Source Measurement page.
Conoscopes are also used to measure scattered light from laser-illuminated surfaces, which is the domain of conoscopic scatterometry. And increasingly, custom conoscopes are being developed to evaluate near-eye displays in AR/VR headsets, one of the fastest-growing applications today.
Conoscopes were developed in the first decade of the 1900s to evaluate liquid crystal displays. Early LCDs were characterized by contrast that decreased with the angle from straight-on and exhibited color-shift at larger angles. These characteristics can easily be quantified with a conoscope.
Our first experience with a conoscope was for measuring the light output from an LCD. Most conoscope requests we receive now are for testing AR/VR devices. However, we still design and manufacture many conoscopes for flat panel display and diffuser measurement. Our custom designs can be tailored to meet your application and achieve any acceptance angle you need.
For a deeper discussion of conoscope optical architecture and the tradeoffs that define practical performance limits, see a Photonics Media article on conoscope design principles , which examines how acceptance angle, NA, sensor sampling, and relay design interact in real instruments.
The stop of a conoscope determines how much of the light source is sampled and the location of the sample. By definition, a stop is the aperture that limits the size of the beam of light passing from a point at the center of the field of view to the image If this subject is of interest, check out our Stops and Pupils page.
The original form of a conoscope had an internal stop. These lenses were used to test LCD’s, and the stop limited the area of the LCD that was being measured. Because the liquid crystal material is located between sheets of glass, it is impossible to place a stop coincident with the area to be measured. Thus, a stop is placed inside the conoscope and projected onto the liquid crystal. The stop and the sample area are images of each other. These conoscopes are still widely used to characterize LCD and OLED panels. They are primarily used to measure contrast and viewing angle.
For applications like scatterometers, DOE and LED measurement, the light source is the stop, so its size and location are the determining factors. Because the light source is external to the conoscope, these applications require an “external stop” design. Measuring the pattern or variation of light with angle is the primary use.
Another form of “external stop” conoscopes is used to test AR/VR headsets. In these conoscopes, a physical stop is placed at the front of the conoscope to mimic the iris of the human eye. This physical stop is a thin piece of metal with a circular hole in the center. Our Modified Conoscopes are a variation on this design and are specifically made for testing AR/VR headsets. Headset MTF, distortion and color uniformity are often measured with these conoscopes. Virtual Image Distance (VID) can also be measured if the pupil of the conoscope is large enough.
Some conoscopes are designed to work at multiple object distances, which is often the case with AR/VR inspection. A conoscope capable of this is called a variable conjugate lens, varifocal, or variable object distance lens. To inspect AR/VR the conoscope must be able to accommodate both near-sighted and far-sighted users. The conoscope must be able to “focus” to adjust for various 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) correspond to an infinite 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.
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 in a conoscope include 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 Back End Lenses make a smaller copy of the intermediate image in the appropriate size and location for the image sensor to capture. 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.
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.
The major function of the Front End Lenses is to perform an angle to position mapping. More specifically, the distance on the intermediate image plane 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 often 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 all conoscope lenses, it is when turning the lens into a scatterometer. For more on that, see our How Conoscopic Scatterometers Work page.
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 has a 50 mm diameter and the image sensor has a 5 mm diameter, we have a magnification of 0.1×. 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 image size for f/#.
These sensor and f/# tradeoffs are among the primary drivers of conoscope pricing, for a full breakdown of what to expect, see our conoscope lens cost guide.
Another option is to decrease the size of the intermediate image. 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! The sample is at the pupil of the Front End Lenses, so the size of the sample is the size of the pupil, so size matters.
Why not leave the sample size the same? The Front End Lenses form a very wide angle lens, so they must operate at a fairly slow f/# to produce a reasonably sharp intermediate 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.
It’s also useful to be aware of several essential design considerations. When specifying a conoscope, the most important constraint 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:
\(\Phi_p \cdot \sin{\Theta} = \Phi_i \cdot \mathrm{NA}_i\)
In this equation, Φp represents the diameter of the entrance pupil, Θ is the half-field of view of the conoscope, Φi 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 (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 NA is defined as the refractive index multiplied by the sine of the marginal ray angle, and the refractive index of air is 1.0, meeting this hypothetical requirement would be an impossibility unless the image sensor is immersed in a high index fluid. It is helpful to keep in mind that lens designs start to become challenging at NA 0.25 (F/2).
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:
\(\Phi_1 = \Phi_p + (2 \cdot 50 \cdot \tan(75^\circ)) = 6 + (2 \cdot 50 \cdot \tan(75^\circ)) = 380\)
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 multi million-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.
\(x = \frac{\Phi_p \cdot \Theta}{OD}\)
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 varifocal range would add to the difficulty.
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:
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.
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.
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. For initial planning, 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.
The default values represent a practical conoscope starting point, 6 mm image height, 50° acceptance angle, 2 mm working distance, 3 mm device diameter. Edit any green field to recalculate specifications for your application. Results update automatically.
*The formulas in this calculator utilize angles in radians for calculation, but allow the input and output in degrees for convenience. If you are using these formulas outside the calculator, be sure to convert your angles to radians!
**This calculator gives approximate values that may change for custom applications. For example, the calculator may output a Φmax = 50mm, but during design there may be a constraint that forces a larger Φmax.
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.
A conoscope is a type of lens that converts an incoming ray angle to a position on the image sensor. Put another way, conoscopes measure the angular distribution of light. This is different from normal lenses…
By Steve Eckhardt Ph.D | June 4th, 2019
The two major constraints on a conoscope for use in measuring displays are the requirement for a pupil in front of the lens and etendue…
By Steve Eckhardt Ph.D | Aug 8th, 2018
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By Steve Eckhardt Ph.D | June 30th, 2025