In the case of aurorae, the emitted light is primarily produced by oxygen atoms in the upper atmosphere that have been disturbed by particles from Earth's magnetotail. These atoms are excited to higher energy states and subsequently relax back to lower energy states by emitting light. Due to the electron spins of each condition being incompatible, ordinary transitions back to the original low energy state are not permitted. However, quantum mechanics allows for alternative pathways, and eventually, the excess energy seeps away, resulting in the familiar green glow of aurorae.

Modulation transfer function (MTF) is one of the most important parameters by which image quality is measured. Optical designers and engineers frequently refer to MTF data, especially in applications where success or failure is contingent on how accurately a particular object is imaged. To truly grasp MTF, it is necessary to first understand the ideas of resolution and contrast, as well as how an object's image is transferred from object to image plane. While initially daunting, understanding and eventually interpreting MTF data is a very powerful tool for any optical designer. With knowledge and experience, MTF can make selecting the appropriate lens a far easier endeavor - despite the multitude of offerings.

MTF testingMastercard

But back to bound electrons. The higher energy electron wave can relax back to a lower energy state by emitting light. The light is, in principle, monochromatic because only discrete electron wave energies exist. In reality, the emitted light is never completely monochromatic. One reason is fundamental, the Heisenberg Uncertainty Principle. The higher the probability of light emission (i.e. the greater the ability to predict the time of its emission) the greater is the light’s frequency spread. Another reason is prosaic. If the atoms are in a high pressure gas, frequent collisions perturb the electron energies and ‘pressure broadening’ spreads the emitted wavelengths.

In traditional system integration (and less crucial applications), the system's performance is roughly estimated using the principle of the weakest link. The principle of the weakest link proposes that a system's resolution is solely limited by the component with the lowest resolution. Although this approach is very useful for quick estimations, it is actually flawed because every component within the system contributes error to the image, yielding poorer image quality than the weakest link alone.

When optical designers attempt to compare the performance of optical systems, a commonly used measure is the modulation transfer function (MTF). MTF is used for components as simple as a spherical singlet lens to those as complex as a multi-element telecentric imaging lens assembly. In order to understand the significance of MTF, consider some general principles and practical examples for defining MTF including its components, importance, and characterization.

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Street lamp tubes have reasonable pressures compared to the upper atmosphere and therefore have slightly radiation broadened light. Astronomer loathed high pressure sodium street lamps have such extensive pressure broadening that their sky polluting light cannot be filtered.

To understand the relation between contrast and image quality, consider an imaging lens with the same resolution as the one in Figure 1 and Figure 4, but used to image an object with a greater line-pair frequency. Figure 5 illustrates that as the spatial frequency of the lines increases, the contrast of the image decreases. This effect is always present when working with imaging lenses of the same resolution. For the image to appear defined, black must be truly black and white truly white, with a minimal amount of grayscale between.

Conceptually, MTF can be difficult to grasp. Perhaps the easiest way to understand this notion of transferring contrast from object to image plane is by examining a real-world example. Figures 8 - 12 compare MTF curves and images for two 25mm fixed focal length imaging lenses: #54-855 Finite Conjugate Micro-Video Lens and #59-871 Compact Fixed Focal Length Lens. Figure 8 shows polychromatic diffraction MTF for these two lenses. Depending upon the testing conditions, both lenses can yield equivalent performance. In this particular example, both are trying to resolve group 2, elements 5 -6 (indicated by the red boxes in Figure 10) and group 3, elements 5 – 6 (indicated by the blue boxes in Figure 10) on a 1951 USAF resolution target (Figure 9). In terms of actual object size, group 2, elements 5 – 6 represent 6.35 – $ 7.13\tfrac{\text{lp}}{\text{mm}} $ (14.03 - 15.75μm) and group 3, elements 5 – 6 represent 12.70 – $ 14.25 \tfrac{\text{lp}}{\text{mm}} $ (7.02 - 7.87μm). For an easy way to calculate resolution given element and group numbers, use our 1951 USAF Resolution Edmund Optics Tech Tool.

Knowledge Center/ Application Notes/ Optics Application Notes/ Introduction to Modulation Transfer Function

The key difference between street lamps and aurorae lies in their degree of monochromaticity. Street lamps, such as high-pressure sodium lamps, have a higher pressure compared to the upper atmosphere. This pressure broadening effect leads to a spread in the emitted wavelengths, causing their light to be less monochromatic. Astronomers often lament the use of high-pressure sodium lamps as their broadened light pollutes the night sky and cannot be easily filtered.

ImageJMTF

When this same principle is applied to the imaging example in Figure 1, the intensity pattern before and after imaging can be seen (Figure 4). Contrast or modulation can then be defined as how faithfully the minimum and maximum intensity values are transferred from object plane to image plane.

In conclusion, the study of aurorae and monochromatic light provides a captivating glimpse into the intricacies of quantum mechanics and atmospheric physics. Understanding the underlying mechanisms behind the production of monochromatic light in aurorae enhances our appreciation for these awe-inspiring natural displays. By exploring the degree of monochromaticity in different light sources, such as street lamps and the Sun, we gain insights into the broader spectrum of light phenomena that grace our skies. So next time you catch a glimpse of the mesmerizing colors dancing across the night sky, take a moment to ponder the intricate interplay between monochromatic light and the atmospheric conditions that create these celestial wonders.

LensMTF

Auroral discharges are from (in Peter’s case) oxygen atoms in the near vacuum of the upper atmosphere disturbed by particles ejected from Earth’s magnetotail. Quantum mechanics rears its head in that the oxygen high energy state is awkward. Ordinary transitions back to the original low lower energy wave are not permitted because the electron spins of each condition are not compatible. But nearly all things happen in quantum mechanics. The electron energy eventually seeps away in the familiar auroral green glow. But ‘eventually’ is comparable to the age of the Universe in atomic terms. The Uncertainty Principle acts again. The time of the emission is grossly uncertain and its HUP pairing with the radiation bandwidth means that the green light has an extremely narrow wavelength spread.

MTF is one of the best tools available to quantify the overall imaging performance of a system in terms of resolution and contrast. As a result, knowing the MTF curves of each imaging lens and camera sensor within a system allows a designer to make the appropriate selection when optimizing for a particular resolution.

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Sunlight has a near continuous spectrum. The original gamma radiation from the core is downgraded by endless energetic collisions on its long journey to the surface. The resulting pressure broadening produces a near thermal equilibrium 'black body' wavelength distribution. Only absorptions by atoms close to the solar surface mar this smooth spectrum.

Modulation transfer function

Under the same testing parameters, it is clear to see that #59-871 (with a better MTF curve) yields better imaging performance compared to #54-855 (Figures 11 – 12). In this real-world example with these particular 1951 USAF elements, a higher modulation value at higher spatial frequencies corresponds to a clearer image; however, this is not always the case. Some lenses are designed to be able to very accurately resolve lower spatial frequencies, and have a very low cut-off frequency (i.e. they cannot resolve higher spatial frequencies). Had the target been group -1, elements 5-6, the two lenses would have produced much more similar images given their modulation values at lower frequencies.

A greater area under the MTF curve does not always indicate the optimal choice. A designer should decide based on the resolution of the application at hand. As previously discussed, an MTF graph plots the percentage of transferred contrast versus the frequency (cycles/mm) of the lines. A few things should be noted about the MTF curves offered by Edmund Optics®:

It is worth noting that sunlight, despite being composed of a wide range of wavelengths, also exhibits pressure broadening due to energetic collisions on its journey from the Sun's core to its surface. However, these broadened wavelengths follow a near thermal equilibrium distribution, leading to a relatively smooth spectrum.

Peter Rosen imaged this at Östersund, Sweden. “We had a beautiful little aurora. There was plenty of fog so we had to find a spot higher up and away from the lake to get an almost free view. Behind us there were sodium streetlights with their typical amber color. The streetlights are the lightsource of the heiligenschein around my camera on the tripod.” Nearly all the light in the scene, except that from the stars, is narrow band atomic radiation. ©Peter Rosen, shown with permission

MTF

"Aurorae and monochromatic light - OPOD". Atmospheric Optics. Accessed on November 23, 2024. https://atoptics.co.uk/blog/aurorae-and-monochromatic-light-opod/.

FOV and focal length

"Aurorae and monochromatic light - OPOD". Atmospheric Optics, https://atoptics.co.uk/blog/aurorae-and-monochromatic-light-opod/. Accessed 23 November, 2024

A practical way of understanding line-pairs is to think of them as pixels on a camera sensor, where a single line-pair corresponds to two pixels (Figure 2). Two camera sensor pixels are needed for each line-pair of resolution: one pixel is dedicated to the red line and the other to the blank space between pixels. Using the aforementioned metaphor, image resolution of the camera can now be specified as equal to twice its pixel size.

Now that the components of the modulation transfer function (MTF), resolution and contrast/modulation, are defined, consider MTF itself. The MTF of a lens, as the name implies, is a measurement of its ability to transfer contrast at a particular resolution from the object to the image. In other words, MTF is a way to incorporate resolution and contrast into a single specification. As line spacing decreases (i.e. the frequency increases) on the test target, it becomes increasingly difficult for the lens to efficiently transfer this decrease in contrast; as result, MTF decreases (Figure 6).

Figure 6 plots the MTF of an aberration-free image with a rectangular pupil. As can be expected, the MTF decreases as the spatial resolution increases. It is important to note that these cases are idealized and that no actual system is completely aberration-free.

On the other hand, the light emitted by oxygen atoms in aurorae has an extremely narrow wavelength spread. The uncertainty in the timing of emission, combined with the Heisenberg Uncertainty Principle's pairing with the radiation bandwidth, results in the green light of aurorae having a remarkably precise wavelength. This narrow spread contributes to the vibrant and distinct colors observed in auroral displays.

For an aberration-free image with a circular pupil, MTF is given by Equation 4, where MTF is a function of spatial resolution $\small{\left( \xi \right)}$, which refers to the smallest line-pair the system can resolve. The cut-off frequency $\small{\left( \xi _c \right)}$ is given by Equation 6.

To properly define the modulation transfer function, it is necessary to first define two terms required to truly characterize image performance: resolution and contrast.

We cannot be sure whether the glow around the camera is a heiligenschein from dew wetted stalks and stubble. It probably is - but it might be an opposition effect!

Aurorae, also known as the Northern and Southern Lights, are captivating displays of light that grace the night skies near the Earth's poles. These natural phenomena have fascinated scientists and skywatchers for centuries. In this article, we will delve into the intriguing connection between aurorae and monochromatic light, shedding light on the underlying physics and mechanisms behind their mesmerizing glow.

The bound electron waves surrounding atoms can have only discrete energies. When an isolated atom is disturbed in some way, for example by a collision with a high energy particle, an electron wave can have its energy increased to another higher energy discrete state. To be pedantic, the higher energy state need not always be discrete for if the energy is high enough we have a free electron and an ionised atom.

Correspondingly, object resolution is calculated using the camera resolution and the primary magnification (PMAG) of the imaging lens (Equations 1 – 2). It is important to note that these equations assume the imaging lens contributes no resolution loss.

When it comes to monochromatic light, we typically think of light that consists of a single wavelength or color. However, in reality, achieving perfect monochromaticity is challenging due to various factors. The Heisenberg Uncertainty Principle, a fundamental concept in quantum mechanics, plays a role in limiting the monochromatic nature of emitted light. The principle states that the more accurately we try to predict the time of light emission, the greater the uncertainty in its frequency or wavelength.

Consider normalizing the intensity of a bar target by assigning a maximum value to the white bars and zero value to the black bars. Plotting these values results in a square wave, from which the notion of contrast can be more easily seen (Figure 3). Mathematically, contrast is calculated with Equation 3:

In addition to the fascinating interplay between monochromatic light and aurorae, there are other optical phenomena that can occur simultaneously. For example, the glow around the camera in photographs of aurorae might be attributed to a heiligenschein, a phenomenon caused by dew-wetted stalks and stubble reflecting light back towards the observer. However, it is essential to consider that this glow could also be an opposition effect, which occurs when sunlight is directly reflected back from small particles, creating a bright spot around the observer's shadow.

Every component within a system has an associated modulation transfer function (MTF) and, as a result, contributes to the overall MTF of the system. This includes the imaging lens, camera sensor, image capture boards, and video cables, for instance. The resulting MTF of the system is the product of all the MTF curves of its components (Figure 7). For instance, a 25mm fixed focal length lens and a 25mm double gauss lens can be compared by evaluating the resulting system performance of both lenses with a Sony monochrome camera. By analyzing the system MTF curve, it is straightforward to determine which combination will yield sufficient performance. In some metrology applications, for example, a certain amount of contrast is required for accurate image edge detection. If the minimum contrast needs to be 35% and the image resolution required is $ 30 \tfrac{\text{lp}}{\text{mm}} $, then the 25mm double gauss lens is the best choice.

In imaging applications, the imaging lens, camera sensor, and illumination play key roles in determining the resulting image contrast. The lens contrast is typically defined in terms of the percentage of the object contrast that is reproduced. The sensor's ability to reproduce contrast is usually specified in terms of decibels (dB) in analog cameras and bits in digital cameras.

Aurorae and monochromatic light - OPOD. Atmospheric Optics. Retrieved from https://atoptics.co.uk/blog/aurorae-and-monochromatic-light-opod/.

A theoretical modulation transfer function (MTF) curve can be generated from the optical prescription of any lens. Although this can be helpful, it does not indicate the actual, real-world performance of the lens after accounting for manufacturing tolerances. Manufacturing tolerances always introduce some performance loss to the original optical design since factors such as geometry and coating deviate slightly from an ideal lens or lens system. For this reason, in our manufacturing sites, Edmund Optics® invests in optical test and measurement equipment for quantifying MTF. This MTF test and measurement equipment allow for the characterization of the actual performance of both designed lenses and commercial lenses (whose optical prescription is not available to the public). As a result, precise integration - previously limited to lenses with known prescriptions - can now include commercial lenses.

Resolution is an imaging system's ability to distinguish object detail. It is often expressed in terms of line-pairs per millimeter (where a line-pair is a sequence of one black line and one white line). This measure of line-pairs per millimeter $ \small{\left(\tfrac{\text{lp}}{\text{mm}}\right)} $ is also known as frequency. The inverse of the frequency yields the spacing in millimeters between two resolved lines. Bar targets with a series of equally spaced, alternating white and black bars (i.e. a 1951 USAF target or a Ronchi ruling) are ideal for testing system performance. For a more detailed explanation of test targets, view Choosing the Correct Test Target. For all imaging optics, when imaging such a pattern, perfect line edges become blurred to a degree (Figure 1). High-resolution images are those which exhibit a large amount of detail as a result of minimal blurring. Conversely, low-resolution images lack fine detail.