The best way to determine an optical system’s resolution is through MTF (Modulation Transfer Function) testing. We want to share from our experience developing multiple testing systems, providing insight on how our testing achieves accurate assessment of the as-built performance of our lenses.
You may be familiar with several ways to quantify lens resolution. For example, reverse projection can potentially give subjective results, while MTF testing provides high precision by comparison. (Check out our resolution testing methods page for more information.)
To measure point source MTF we observe a point that is sufficiently small and use a Fourier transform to get the frequency information from the resulting image. (We have a great page dedicated to MTF if you would like to learn more.) The Fourier transform is widely used in many other applications such as signal processing or vibration analysis, so it may already be familiar to you.
We prefer point source MTF to test our lenses as it allows us to analyze discrete field points. Not to be understated, another crucial benefit to MTF testing is the potential to reveal misalignment. A misaligned lens is much more apparent when looking at the image of a point source than a slanted edge target or AF1951 resolution test target. (For more on Optical Aberrations visit our page here.).
The MTF performance of our lenses is often a key factor in determining if our product meets expectations. We have the privilege of designing a wide variety of lenses which necessitates that our MTF tester(s) accommodates each lens respectively.
The point source must be selected to match the LUT specifications in several ways to provide accurate MTF measurements.
The first consideration is the object distance the lens is designed to work at. Many imaging lenses are specified to work at an infinite object. Fortunately, there are many collimators capable of projecting a point at infinity. However, the size of a large collimator can be expensive. The exit pupil of the collimator must be greater than or equal to the entrance pupil of the LUT or it will effectively stop down (increase the F/#) of the LUT and change the test conditions.
In addition, the collimator must be diffraction-limited over the entrance pupil of the LUT. If a finite object distance is required, a pinhole can be placed at the desired distance. We often prefer more flexibility and use a focusable point source projector that can easily be adjusted for different object distances.
Determining the pinhole size is another critical factor as it acts as a reference for a perfect lens. If the pinhole is too small, not enough light will be able to get to the image sensor, and if the pinhole is too large it will not provide a diffraction limited source. The diffraction limit is the limit of what is resolvable based on the diffraction of the light and is determined by the Airy disk diameter.
This is given by D=2.44*λ*(f/#)
Where D is the airy disk diameter, λ is the wavelength, and f/# is the f-number of the projected point, calculated by dividing the focal length of the point source projector by the entrance pupil of the LUT.
We generally choose a pinhole that is less than half of the Airy disk diameter to make sure that it is diffraction limited. We typically do not go below 1/3 of the Airy disk diameter as this may result in a point so dim that stray light and noise compromise the measurement.
Finally, the light source requires careful consideration. LEDs are available in a wide range of colors and provide the brightness needed for MTF testing. Depending on the wavelength specification for MTF, an appropriate LED can usually be chosen that closely matches the desired spectrum. Beware that with non-visible light, optical surfaces of the point source projector must have coatings that ensure acceptable transmission at the specified wavelengths.
We have designed and built a custom point source projector to satisfy the requirements of testing at a wide range of object distances (-10 to +10 diopters) and lens entrance pupils up to 10mm. This is especially useful for testing conoscopes and other custom lenses, which often have small entrance pupils and require precise control of object distance when evaluating AR and VR optical systems.
The point source projector is designed to be able to easily interchange pinholes and light sources, depending on the specifications of the LUT. It is designed to provide a diffraction limited point source for visible and NIR wavebands and is coated to ensure transmission from 430nm to 1000nm. This allows us to keep blue, red, green, white, and NIR light sources on hand for testing in common wavebands.
Additionally, when our point source projector does not accommodate a lens, we use a variety of other projector setups for testing. For example, for large aperture lenses with objects at infinity we use a telescope to project a diffraction limited point source into the lens.
The image sensor has a similar requirement to the pinhole. It must have a small enough pixel size to be able to see if there are problems with the lens. The Nyquist frequency determines the appropriate pixel size, as discrete samples of the point must be taken.
Where vN is the Nyquist frequency in cycles/mm, and Px is the pixel size in µm.
As the resolution of the lens being tested approaches the Nyquist frequency of the image sensor, the calculated contrast starts to drop. It is affected by whether the point lands on the center of a pixel or on the edge between pixels. Correction factors can be used when approaching Nyquist frequency to address the resolution drop off. However, these are average values so they do not control for where on the pixel the point is landing.
Consequently, a better MTF value is sometimes reported than the actual MTF. This should be avoided because the goal is to ensure that the lenses have performance that will be acceptable for the final application.
Another way to achieve the sensor resolution requirement is to magnify the image formed by the LUT. This adds complexity to the setup, so we prefer to use an image sensor with sufficiently small pixels. This necessitates leaving some leeway between the frequency that is being tested and the pixel Nyquist frequency.
An additional requirement for the image sensor is perpendicularity to the optical axis. This is crucial for fast lenses (low F/#) where as little as 10µm of de-focus can have a significant detrimental effect on the performance. Since image sensors with small pixels are generally smaller in overall size than those designed for the final application, the image sensor in our MTF tester must also be translated to measure all required field points.
The kinematics of the MTF tester must be designed to adapt to the characteristics of the lens under test (LUT). Most camera lenses are designed to image an object that is perpendicular to the optical axis. For finite conjugate systems, this means that as the point source moves laterally to different object points, its angle must also be adjusted to ensure it remains directed toward the center of the entrance pupil of the lens.
However, many of our conoscopes are designed with an object surface that is a sphere centered at the entrance pupil of the lens. In this case, the point source is rotated about the entrance pupil of the lens. We can have interchangeable setups when the object is positioned at infinity as both configurations become equivalent.
Whether the object surface is spherical or planer, many MTF testers take advantage of the rotational symmetry of a lens and rotate the lens to test other azimuths. When the lens is not rotationally symmetric other problems arise, and a more complex structure is needed as described by Hu et. Al.
We use our custom point source projector mounted to a gimbal to test asymmetric lenses as it allows us to focus through a wide range of object distances.
Hu, Yuan & Cheng, Dewen & Wang, Yongtian & Peng, Haichao. (2015). Effective modulation transfer function measurement method for an off-axis optical system. Applied Optics. 54. 7471. 10.1364/AO.54.007471.
Widely available MTF analysis software tends to analyze slanted edge MTF. The advantages of using point source MTF underline the potential of custom software. Steve Eckhardt has many years of experience writing optical software which has enabled more flexibility in setting up the tests for each custom lens.
Our software is designed to give both accurate analysis of the point, and easy troubleshooting of the optical system. For example, the exposure of the camera needs to be controlled for accurate analysis. This cannot be done in post processing of the image as there will be some information lost. The most accurate MTF is found when the brightest pixel is just under the maximum brightness. If it is overexposed some contrast is lost, and if it is not bright enough, some of the light spread around the point will not be visible.
To facilitate this, we have a cross section of the brightness of the point. For the MTF analysis, the software must know the pixel size of the camera and the wavelength of the point source. Our software affords us a few more benefits including using a calibration factor to map the field points to the right image sensor location. Additionally, we can pull the test specifications and field test points from a file for each lens, ensuring consistent ease of use.
Eckhardt Optics’ Point Source MTF tester has been validated by comparing the results for a given lens to the MTF measured with a TRIOPTICS ImageMaster MTF tester for the same lens.
To more easily and efficiently test all desired field points we use two types of motorized MTF testers. Otherwise, manual set ups are used for lenses outside of our systems capabilities.
Our swing arm tester is a motorized version of a manual swing arm MTF tester. This enables us to both test field angles out to 90° and rotate the lens to evaluate performance at all azimuths.
We can use any point source that we want as they are simply mounted to a rail on the optical bench. The point source projector can also be used for variable conjugate testing, or if we need to test a LUT with a large aperture, a large telescope can also be mounted as a point source.
We designed a periscope lens tester for lenses that are not axially symmetric or are simply difficult to rotate. It uses a gimble to change the field angle and azimuth of the light source and then moves the image sensor in both directions to capture the result. Currently, this tester only uses our point source projector since it is designed to mount to the gimbal, so it is limited to 10mm entrance pupils.
In addition to our motorized MTF testing systems we use custom manual setups to test a variety of lenses. These set ups often use a swing arm on a precision rotation stage, and we employ the optomechanics we have on hand. It is very rare that a lens cannot be tested with a custom setup. However, as the complexity increases, more time is spent on setup, and more custom components are likely to be needed.
As always, we’re interested in finding creative ways to solve complicated (or expensive) problems. If we can help you with your next project, let us know!