Far away from the aperture, the angle at which the first minimum occurs, measured from the direction of incoming light, is given by

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Light from a uniformly illuminated circular aperture (or from a uniform, flattop beam) will exhibit an Airy diffraction pattern far away from the aperture due to Fraunhofer diffraction (far-field diffraction).

where J0 and J1 are Bessel functions. Hence the fractions of the total power contained within the first, second, and third dark rings (where J1(kasinθ) = 0) are 83.8%, 91.0%, and 93.8% respectively.[5]

where x is the separation of the images of the two objects on the film and f is the distance from the lens to the film. If we take the distance from the lens to the film to be approximately equal to the focal length of the lens, we find

Camera test

Fill the frame with your blank target area, manually dial the lens completely out of focus, and snap off a frame. When you open the image on your computer, look for dark spots. See any? Those are your culprits.

Eradicating any pest requires knowledge of where it’s hiding and how it got there, so I want to start by explaining why sensor dust is such a common problem.

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In a camera or imaging system an object far away gets imaged onto the film or detector plane by the objective lens, and the far field diffraction pattern is observed at the detector. The resulting image is a convolution of the ideal image with the Airy diffraction pattern due to diffraction from the iris aperture or due to the finite size of the lens. This leads to the finite resolution of a lens system described above.

While it might not seem like it, the world in which we live is full of dust. Regardless of how clean your studio is or how lint-free your camera bags are, microscopic dust particles will settle on your camera at some point.

In the example below, certain spots were visible in the original photo. By inverting the image, several additional trouble spots on my sensor became apparent:

The intensity I0 at the center of the diffraction pattern is related to the total power P0 incident on the aperture by[4]

where R is the distance from the aperture. The half maximum of the central Airy disk (where J1(x) = 1 / 2) occurs at x = 1.61633...; the 1/e2 point (where J1(x) = 1 / e2) occurs at x = 2.58383..., and the maximum of the first ring occurs at x = 5.13562....

In practice, the conditions for uniform illumination can be met by placing the source of the illumination far from the aperture. If the conditions for far field are not met (for example if the aperture is large), the far-field Airy diffraction pattern can also be obtained on a screen much closer to the aperture by using a lens right after the aperture (or the lens itself can form the aperture). The Airy pattern will then be formed at the focus of the lens rather than at infinity.

And that’s before considering outdoor use or changing lenses. When using budget zoom lenses, the simple act of repeatedly zooming in and out can “inhale” dust particles into the lens, which can then work their way into your camera over time. Once inside, they can settle on the mirror or sensor.

Camera sensor

However, if you don’t see any dust spots when reviewing your images, they may still be present – just hidden. If you typically shoot at wide apertures, most dust particles will not show up, even against bright backgrounds. Dust may also be less noticeable in images with a lot of detail, but that doesn’t mean it’s gone.

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A circular laser beam with uniform intensity profile, focused by a lens, will form an Airy pattern at the focal plane of the disk. The intensity at the center of the focus will be I0,Airy = (P0A) / (λ2f2) where P0 is the total power of the beam, A = πD2 / 4 is the area of the beam (D is the beam diameter), λ is the wavelength, and f is the focal length of the lens.

The expression for I(θ) above can be integrated to give the total power contained in the diffraction pattern within a circle of given size:

A circular laser beam with uniform intensity across the circle (a flattop beam) focused by a lens will form an Airy disk pattern at the focus. The size of the Airy disk determines the laser intensity at the focus.

Additionally, changing lenses significantly increases the likelihood of dust finding its way to your sensor. (And if you change lenses in a dirty environment, the problem is compounded!)

On DSLRs, sensor dust won’t show itself until it leaves dark spots of varying sizes on your photos. If you can see dust when looking through your DSLR viewfinder, it’s not on the sensor.

where Ε is the source strength per unit area at the aperture, A is the area of the aperture (A = πa2) and R is the distance from the aperture. At the focal plane of a lens, I0 = (P0A) / (λ2f2). The intensity at the maximum of the first ring is about 1.75% of the intensity at the center of the Airy disk.

is a commercial/portrait photographer based in Atlanta, GA. Still an avid street photographer and film shooter, Jeff also launched a kids photography class called: Digital Photo Challenges.

If you do find dust spots that you want to remove, there are many do-it-yourself sensor-cleaning options available. If you are comfortable doing this task on your own, great. If not – and if you aren’t, I don’t blame you – local camera shops provide sensor-cleaning services for a nominal fee.

Lightroom also offers a handy way to look for dust spots. Simply open your test image in the Develop module, then select the Remove tool in the upper right-hand corner (under the histogram). Check the Visualize Spots box, and Lightroom will create a high-contrast monochrome mask, which you can use to quickly identify any problematic areas!

Sensor dust can generally be fixed in Lightroom or Photoshop without too much hassle, but why spend extra time in front of the computer when you don’t have to? If you strive to get things right in the camera (and you should), this is absolutely something to keep on a semi-regular checklist.

where I0 is the intensity in the center of the diffraction pattern, J1 is the Bessel function of the first kind of order one, k = 2π / λ is the wavenumber, a is the radius of the aperture, and θ is the angle of observation, i.e. the angle between the axis of the circular aperture and the line between aperture center and observation point. , where q is the radial distance from the optics axis in the observation (or focal) plane and N = R / d (d=2a is the aperture diameter, R is the observation distance) is the f-number of the system. If a lens after the aperture is used, the Airy pattern forms at the focal plane of the lens, where R = f (f is the focal length of the lens). Note that the limit for (or for ) is I(0) = I0.

Similar equations can also be derived for the obscured Airy diffraction pattern[6][7], which is the diffraction pattern from an annular aperture or beam, i.e. a uniform circular aperture (beam) obscured by a circular block at the center.

No test is perfect, and this one is no different. However, there is a neat trick you can run in Photoshop as an added layer of detection. By holding down Ctrl/Cmd + I, you will invert the image, creating a negative. The dark spots (if any) will now appear white against a dark background, making them easier to see.

If a dust spot is visible upon a quick glance at the image, you’re likely looking at a pretty bad spot on your sensor – one that won’t leave just because you squeeze a little air at it. You may need a thorough sensor cleaning to get rid of your worst offenders.

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where ε is the annular aperture obscuration ratio, or the ratio of the diameter of the obscuring disk and the diameter of the aperture (beam). , and x is defined as above: where R is the radial distance in the focal plane from the optical axis, λ is the wavelength and N is the f-number of the system. The encircled energy (the fraction of the total energy contained within a circle of radius R centered at the optical axis in the focal plane) is then given by:

Have you experienced problems with sensor dust? Do you have a good way of dealing with it? Share your thoughts in the comments below!

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The smallest f-number for the human eye is about 2.1.[3] The resulting resolution is about 1 μm. This happens to be about the distance between optically sensitive cells, photoreceptors, in the human eye.[citation needed]

but is the f-number of a lens. A typical setting for use on an overcast day would be f/8.[1] For blue visible light, the wavelength λ is about 420 nanometers.[2] This gives a value for x of about 0.004 mm. In a digital camera, making the pixels of the image sensor smaller than this would not actually increase image resolution.

You can run this against a clear sky, a white sheet of paper, or even your computer screen. Start by switching your camera into Aperture Priority mode. Make sure you use Matrix or Evaluative metering and the lowest possible ISO.

where λ is the wavelength of the light and d is the diameter of the aperture. The Rayleigh criterion for barely resolving two objects is that the center of the Airy disc for the first object occurs at the first minimum of the Airy disc of the second. This means that the angular resolution of a diffraction limited system is given by the same formula.

The diffraction pattern resulting from a uniformly illuminated circular aperture has a bright region in the center, known as the Airy disc which together with a series of concentric rings is called the Airy pattern (both named after George Airy). The diameter of this disc is related to the wavelength of the illuminating light and the size of the circular aperture.

The first effective way to identify sensor dust spots is when reviewing your images. You can do this on your camera LCD, but it’s generally better to review for dust on a large monitor.

Which leads to a question: How do you know if your camera sensor does indeed have dust? How can you tell the difference between dust on the mirror, dust on the viewfinder, or dust on the sensor? In this article, I share several strategies to help you check for sensor dust – so that you can keep your camera sensor (and your photos!) dust-free!

We all know that dust on a DSLR or mirrorless camera sensor can range from mildly annoying to downright problematic. You can try avoiding it, but the unfortunate truth about digital photography is that, regardless of how careful you are, dust will eventually find its way onto your sensor.

The most important application of this concept is in cameras and telescopes. Due to diffraction, the smallest point to which one can focus a beam of light using a lens is the size of the Airy disk. Even if one were able to make a perfect lens, there is still a limit to the resolution of an image created by this lens. An optical system in which the resolution is no longer limited by imperfections in the lenses but only by diffraction is said to be diffraction limited.

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On mirrorless cameras, sensor dust can show itself in the electronic viewfinder. However, it’s generally easier to see when reviewing your images on a computer monitor, as I explain in the next section:

So while there are steps you can take to minimize dust (holding the camera with the lens mount facing down while changing lenses, not changing lenses outside, etc), it’s inevitable that dust will find your camera sensor eventually. That’s why, in addition to minimizing dust exposure, it’s important that you regularly check for dust!

Therefore the smallest angular separation two objects can have before they significantly blur together is given as stated above by

I suppose you could prevent any sensor dust by putting a prime lens on your camera when it’s brand new and never, ever taking it off. But since that’s almost certainly not going to happen, dealing with dust is a critical part of regular camera maintenance.

Say that you’re worried about dust, but you’re not sure whether your sensor has a significant dust problem. That’s when I recommend doing a simple test.

The Airy disc (or Airy disk) is a phenomenon in optics. Owing to the wave nature of light, light passing through an aperture is diffracted and forms a pattern of light and dark regions on a screen some distance away from the aperture (see interference).

Whether you pay to have your sensor professionally cleaned or you are comfortable enough doing it yourself, it’s possible to overdo it. Sensors have a delicate coating that can be damaged by over-cleaning, and frequent cleaning increases the odds of damaging your sensor. So how do you do a quick and easy evaluation to see if your sensor really needs cleaning?

The conditions for being in the far field and exhibiting an Airy pattern are: the incoming light illuminating the aperture is a plane wave (no phase variation across the aperture), the intensity is constant over the area of the aperture, and the distance R from the aperture where the diffracted light is observed (the screen distance) is large compared the aperture size, and the radius a of the aperture is not too much larger than the wavelength λ of the light. The last two conditions can be formally written as R > a2 / λ .

You can recognize a dust spot if it appears in the same place in multiple images, particularly in those taken at apertures such as f/8 or smaller. Dust spots are especially visible when positioned against the sky.

It’s important to recognize that while dust can be visible if it lands on the mirror or the viewfinder, neither of these areas has any effect on your photos. While the viewfinder and the mirror are important for viewing the scene and taking the photo, mirror and viewfinder dust will have absolutely no impact on your images. You can often see these types of dust with the naked eye when you look through the viewfinder. They can be annoying, but they’re often fixed with an air blower.

A Gaussian beam with 1 / e2 diameter of D focused through an aperture of diameter D will have a focal profile that is nearly Gaussian, and the intensity at the center of the focus will be 0.924 times I0,Airy.[7]

Either way, by knowing how to quickly identify the problem, you’re in a much better position to address it and get back to taking clean, crisp, dust-free photos.

If two objects imaged by a camera are separated by an angle small enough that their Airy disks on the camera detector start overlapping, the objects can not be clearly separated any more in the image, and they start blurring together. Two objects are said to be just resolved when the maximum of the first Airy pattern falls on top of the first minimum of the second Airy pattern (the Rayleigh criterion).