This typically changes with the size at which you print a picture -- bigger pictures are normally viewed from a greater distance, so more blur is acceptable. Most lens markings, etc., are defined based on a print around 8x10 being viewed at roughly arm's length distance (a couple of feet or so). The math for this works out fairly simple: start with an estimate of visual acuity, which will be measured as an angle. Then you just figure out what size that angle works out to at a specified distance.

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At a narrower aperture, the diaphragm DOES block some light from the periphery of the light cone, while light from the center is allowed through. The maximum angle of light rays reaching the sensor is low (less oblique). This causes the maximum CoC to be smaller, and progression from a focused point of light to maximum CoC is slower. (In an effort to keep the diagram as simple as possible, the effect of spherical aberration was ignored, so the diagram is not 100% accurate, but should still demonstrate the point):

If we ignore print size and film, for a given digital sensor with a specific pixel density, DoF is a function of focal length, relative aperture, and subject distance. From that, one could make the argument that DoF is purely a function of the lens, as "subject distance" refers to the distance at which the lens is focused, which would also be a function of the lens.

The term circle of confusion is applied more generally, to the size of the out-of-focus spot to which a lens images an object point. It relates to 1. visual acuity, 2. viewing conditions, and 3. enlargement from the original image to the final image. In photography, the circle of confusion (CoC) is used to mathematically determine the depth of field, the part of an image that is acceptably sharp.

The thermal expansion coefficients of materials vary significantly and influence structural durability and performance in industries like electronics and construction. The material coefficient of thermal expansion for steel, for example, is determined by how much the structure's dimensions change in response to temperature variations.

DOF simply tells the photographer at what distances before and aft of the focus distance that blurriness will occur. It does not specify how blurry or what “quality” those areas will be. The design of the lens, the design of the diaphragm, and your background define the characteristics of the blur—its intensity, texture, and quality.

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The expansion is quantified by using the coefficient of linear expansion, a formula representing the fractional change in length per unit temperature change. The coefficient of linear expansion looks as follows:

Q. Can lenses be designed to give more depth of field for the same aperture and focal length? A. Given the math, I would have to say no. I am not an optical engineer, so take what I say here with the necessary grain of salt. I tend to follow the math, though, which is pretty clear about depth of field.

Hyperfocal distance is defined as the distance, when the lens is focused at infinity, where objects from half of this distance to infinity will be in focus for a particular lens. Alternatively, hyperfocal distance may refer to the closest distance that a lens can be focused for a given aperture while objects at a distance (infinity) will remain sharp.

It's because of the crop factor and the resulting focal length along with the necessary aperture for the light gathering ability of the sensor that gives the greatest affect upon your calculations.

Unlike DoF for moderate to large subject distances, with 1:1 (or better) macro photography, you are ALWAYS enlarging for print, even if you print at 2x3". At common print sizes such as 8x10, 13x19, etc., the factor of enlargement can be considerable. One should assume CoC is at the minimum resolvable for your imaging medium, which is still likely not small enough to compensate for apparent DoF shrink due to enlargement.

CoC (mm) = viewing distance (cm) / desired final-image resolution (lp/mm) for a 25 cm viewing distance / enlargement / 25

Q. Is it just a property of the lens? A. Simply put, no, although if you ignore CoC, one could (given the math) make the argument that it is. Depth of field is a "fuzzy" thing, and depends a lot on viewing context. By that, I mean it depends on how large the final image being viewed is in relation to the native resolution of the sensor; the visual acuity of the viewer; the aperture used when taking the shot; the distance to subject when taking the shot.

Thermalexpansion of gases

Complex mathematics aside, DoF can be intuitively visualized with a basic understanding of light, how optics bend light, and what effect the aperture has on light.

There are several mathematical formulas that can be used to calculate the depth of field. Sadly, there does not seem to be a single formula that accurately produces a depth of field at any distance to subject. Hyperfocal Distance, or the distance where you effectively get maximum DoF, can be calculated as so:

Thermalexpansion examples

It is vital for engineers to predict how materials may behave under various temperature fluctuations correctly. This prediction offers critical insights into material choice and structural integrity design considerations. It allows engineers to accommodate these dimensional changes in CTE materials. Materials with lower CTE offer decent dimensional stability. Higher CTE indicates more contraction or expansion in a substrate, which aligns with use in structures that need more flexibility to adjust to temperature changes.

Assuming we pick one number for that and stick to it, depth of field only depends on two factors: the aperture and the reproduction ratio. The larger the reproduction ratio (i.e., the larger an item appears on the sensor/film compared to its size in real life) the less depth of field you get. Likewise, the larger the aperture (larger diameter opening -- smaller f/stop number) the less depth of field you get.

Ok for a change I'm going to dispense with the formulas, photos of rulers and definitions of "magnification" and go with what you actually experience in practice. The major factors that actually matter to shooting are:

A higher resolution sensor and a better quality lens will produce better bokeh but even a cellphone sized sensor and lens can produce reasonably acceptable bokeh.

Q. Does it change with print size? A. Given the answer to the previous question, possibly. Scaling an image above, or even below, its "native" print size can affect what value you use for the minimum acceptable CoC. Therefor, yes, the size(es) you intend to print at do play a role, however I would say the role is generally minor unless you print at very large sizes.

\$H\$ is the hyperfocal distance; \$f\$ is the lens's focal length; \$N\$ is the f-number (relative aperture) of the lens; and \$c\$ is the circle of confusion (CoC) diameter.

Coefficient ofthermalexpansion formula

As with everything, one should always prove the concept by actually running the math. Here are some intriguing results when running the formulas above with F# code in the F# Interactive command line utility (easy for anyone to download and double check):

Specifically, a photographic aperture (nowadays) is universally measured as a fraction of the focal length -- it's written like a fraction (f/number) because that's what it is.

Coefficient ofthermalexpansion table PDF

The original (longer) answer is here - this is the abridged version. Simply making a one sentence answer with a link causes the answer to be converted to a comment to the above question, with a risk of deletion because it's a comment.

Circle of confusion: In idealized ray optics rays are assumed to converge to a point when perfectly focused, the shape of a defocus blur spot from a lens with a circular aperture is a hard-edged circle of light. A more general blur spot has soft edges due to diffraction and aberrations (Stokseth 1969, paywall; Merklinger 1992, accessible), and may be non-circular due to the aperture shape.

$$ \begin{align} D_\text{n} &= \frac{Hs}{H + s} \\ D_\text{f} &= \frac{Hs}{H - s} && \text{for }s

Check out this article "Staging Foregrounds" by R.J. Kern on foreground blur, which includes many photos with background and foreground blur.

Bokeh: The quality of the blurring of the out of focus areas of the image outside of the depth of field when the lens is correctly focused on the subject.

Cte thermalcalculator

Linear thermal expansion refers to a material's capacity to expand in length when subjected to an increase in temperature. This happens because the kinetic energy of the material's molecules surges, leading to a vibration that creates more space between them. Conversely, a reduction in temperature, which reduces vibrational energy, causes the material to contract.

Depth of field: The distance between the nearest and farthest objects in a scene that appear acceptably sharp in an image. Although a lens can precisely focus at only one distance at a time, the decrease in sharpness is gradual on each side of the focused distance, so that within the DOF, the unsharpness is imperceptible under normal viewing conditions.

For example, even a really fast (large aperture) lens that has a short focal length makes it fairly difficult to high reproduction ratio. For example, if you take a picture of a person with a 20mm f/2 lens, the lens has to practically touch them before you get a very large reproduction ratio. At the opposite extreme, longer lenses often appear to have less depth of field because they make it relatively easy to achieve a large reproduction ratio.

In the average case, one can assume that CoC is always the minimum achievable with a digital sensor, which these days rolls in at an average of 0.021mm, although a realistic range covering APS-C, APS-H, and Full Frame sensors covers anywhere from 0.015mm – 0.029mm. For most common print sizes, around 13x19" or lower, an acceptable CoC is about 0.05mm, or about twice the average for digital sensors. If you are the type who likes to print at very large sizes, CoC could be a factor (requiring less than 0.01mm), and your apparent DoF in a big enlargement will be smaller than you calculate mathematically.

Modulation Transfer Function (MTF) or Spatial Frequency Response (SFR): The relative amplitude response of an imaging system as a function of input spatial frequency. ISO 12233:2017 specifies methods for measuring the resolution and the SFR of electronic still-picture cameras. Line pairs per millimeter (lp/mm) was the most common spatial frequency unit for film, but cycles/pixel (C/P) and line widths/picture height (LW/PH) are more convenient for digital sensors.

@Matt Grum's comment is quite good: you do have to be really careful to specify conditions, or you can end up with three people saying things that seem to conflict, but are really just talking about different conditions.

For example, it's pretty well known that at f/1.4 you'll get less depth of field than at f/2.8. What may not immediately be so obvious is that (for example) a 50 mm f/1.4 lens and a 100 mm f/2.8 lens have the same effective diameter. It's the wider angle at which light rays enter the 50 mm lens that gives it less depth of field than the 100 mm lens, even though the two have exactly the same physical diameter.

Since the final-image size is not usually known at the time of taking a photograph, it is common to assume a standard size such as 25 cm width, along with a conventional final-image CoC of 0.2 mm, which is 1/1250 of the image width. Conventions in terms of the diagonal measure are also commonly used. The DoF computed using these conventions will need to be adjusted if the original image is cropped before enlarging to the final image size, or if the size and viewing assumptions are altered.

Subject distance. This is a really important consideration. Depth of field gets drastically shallower when you start to get really close. This is important as at macro focussing distances DoF is a major problem. It also means you can get shallow DoF regardless of aperture if you get close enough, and that if you want deep DoF in low light compose to focus further away.

It's the aperture advantage that makes the full frame sensor a better and more expensive choice both for camera and lenses and often for features (FPS not being one of them, nor size and weight).

At relatively close distances, the DOF is nearly symmetrical, with about half of the focus area existing before the focus plane and half appearing after. The farther the focal plane moves from the image plane, the larger the shift in symmetry favoring the area beyond the focal plane. Eventually, the lens focuses at the infinity point and the DOF is at its maximum dissymmetry, with the vast majority of the focused area being beyond the plane of focus to infinity. This distance is known as the “hyperfocal distance” and leads us to our next section.

This is an excellent question, and one that has different answers depending on context. You mentioned several specific questions each of which might warrant their own answers. I'll try to address them more as a unified whole here.

Going to a medium sized sensor over a tiny sensor further advantages the larger sensor but bokeh likely isn't the best use case to justify 20x+ times price difference.

First, to define DoF meaningfully, you need to specify the amount of "blur" you're willing to accept as sufficiently sharp. Depth of field is basically just measuring when something that started as a point in the original will be blurred enough to become larger than whatever size you've picked out.

The formula is fairly simple, outside of the pupil magnification aspect. A true, properly built macro lens will have largely equivalent entrance and exit pupils (the size of the aperture as viewed through the front of the lens (entrance) and the size of the aperture as viewed from the back of the lens (exit)), although they may not be exactly identical. In such cases, one can assume a value of \$P = 1\$, unless you have reasonable doubt.

Every amateur magazine (and ezine now) loves to say 'switch to a wide angle lens for more depth of field'... but if you keep the subject the same size in the frame (by moving in closer) then the sharp bits have the same limits. Walking backwards with the lens you've got on will give more DOF too, but maybe you like the shot the way it is already set up?

There are several questions here about the definition of depth of field, about focal length, and about subject distance. And of course there's the basic how does aperture affect my photographs. And plenty of how do I get super-shallow d.o.f questions. There's related questions like this one. But there's no be-all-end-all question asking:

What you will see are more gradual cut-offs in sharpness so that the background & foreground appear sharper (not sharp as if within the DOF!) hence the lovely out of focus backgrounds with long lenses and the nearly sharp ones with wide angles.

Switching lenses or changing subject-to-camera in accordance with the crop factor when switching between an APS-C and full frame camera to maintain identical framing results in a similar DOF. Moving your position to maintain identical framing slightly favors the full frame sensor (for a greater DOF), it's only when changing lenses to match the crop factor and maintain framing that the larger sensor gains a narrower DOF (and not by much).

The following answer was originally published (by me) as an answer about background bokeh but it necessarily explains depth of field, with a bias to explaining fore and background blur.

Criteria relating CoC to the lens focal length have also been used. Kodak (1972), 5) recommended 2 minutes of arc (the Snellen criterion of 30 cycles/degree for normal vision) for critical viewing, giving CoC ≈ f /1720, where f is the lens focal length. For a 50 mm lens on full-frame 35 mm format, this gave CoC ≈ 0.0291 mm. This criterion evidently assumed that a final image would be viewed at “perspective-correct” distance (i.e., the angle of view would be the same as that of the original image):

Using the same focal length lens on an APS-C and full frame camera at the same subject-to-camera distance produces two different image framings and causes the DOF distance and thickness (depth, of the field) to differ.

Cte thermalformula

This COC value represents the maximum blur spot diameter, measured at the image plane, which looks to be in focus. A spot with a diameter smaller than this COC value will appear as a point of light and, therefore, in focus in the image. Spots with a greater diameter will appear blurry to the observer.

All the other factors (sensor size and focal length being the two more obvious) only affect depth of field to the extent that they affect the reproduction ratio or the aperture.

The greater number of pixels per dot of light certainly will produce smoother bokeh but so would moving closer with a small sensor camera. You can charge proportionality more for use of more expensive equipment if you make money off of your photos or videos, otherwise a bit of footwork or additional lower cost lenses will save you a lot of money over investing in a larger format system.

Q. Does it change with camera sensor size? A. Ultimately, it depends here. More important than the size of the sensor would be the minimum Circle of Confusion (CoC) of the imaging medium. Curiously, the Circle of Confusion of an imaging medium is not necessarily an intrinsic trait, as the minimum acceptable CoC is often determined by the maximum size you intend to print at. Digital sensors do have a fixed minimum size for CoC, as the size of a single sensel is as small as any single point of light can get (in a Bayer sensor, the size of a quartet of sensels is actually the smallest resolution.)

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Is it just a property of the lens? Can lenses be designed to give more depth of field for the same aperture and focal length? Does it change with camera sensor size? Does it change with print size? How do those last two relate?

Photography: In photography the sensor size is measured based upon the width of film or the active area of a digital sensor. The name 35 mm originates with the total width of the 135 film, the perforated cartridge film which was the primary medium of the format prior to the invention of the full frame DSLR. The term 135 format remains in use. In digital photography, the format has come to be known as full frame. While the actual size of the usable area of photographic 35 mm film is 24w×36h mm the 35 millimeters refers to the dimension 24 mm plus the sprocket holes (used to advance the film).

Focal length. This does affect depth of field, but only in certain ranges, when maintaining subject size. Wide lenses have very deep depth of field at most subject distances. Once you get past a certain point, DoF changes very little with focal length. This is important again because if you want to increase / decrease DoF you can use focal length to do this whilst still filling the frame with your subject.

Mathematically, it is clear why DoF is not simply a function of the lens, and involves either the imaging medium or print size from a CoS perspective. To clearly specify the factors of DoF:

Cte thermalunits

Edit2: Since I (sort of) persuaded @jrista to remove his diagram relating focal length to depth of field, I should probably try to explain why there's not a relationship between focal length and depth of field -- at least when you look at things the way they're normally measured in photography.

On the other hand, if you change the focal length but maintain the same photographic aperture (f/stop), the depth of field also remains constant because as the focal length increases the diameter increases proportionally so the rays of light are getting focused on the film/sensor from the same angles.

Sensor size. This affects DoF when you maintain the same subject distance and field of view between sensor sizes. The bigger the sensor the shallower the depth of field. DSLRs have much bigger sensors than compacts, and so for the same FoV and f-ratio they have shallower DoF. This is important because by the same token cropping images increases DoF when maintaining the same final output size, as it's akin to using a smaller sensor.

The output of the above program is intriguing, as it indicates that depth of field is indeed directly influenced by focal length as an independent factor from relative aperture, assuming only focal length changes and everything else remains equal. The two DoF's converge at f/1.4 and f/5.6, as demonstrated by the above program:

The term 'infinity' here is not used in its classical sense, rather it is more of an optical engineering term meaning a focal point beyond the hyperfocal distance. The full formula for calculating DoF directly, without first calculating hyperfocal distance, as as follows (substitute for \$H\$):

The hyperfocal distance is variable and a function of the aperture, focal length, and aforementioned COC. The smaller you make the lens aperture, the closer to the lens the hyperfocal distance becomes. Hyperfocal distance is used in the calculations used to compute DOF.

Intriguing results, if a little non-intuitive. Another convergence occurs when the distances are adjusted, which provides a more intuitive correlation:

Subject: The object that you intend to capture an image of, not necessarily everything that appears in frame, certainly not Photo Bombers, and often not objects appearing in the extreme fore and backgrounds; thus the use of bokeh or DOF to defocus objects which are not the subject.

There are only two factors that actually affect DOF - aperture and magnification - yes switching distance, sensor size, focal length, display size, and viewing distance appear to have an effect but they are all just changes in the size of the image (the subject/part-you're-looking at) as seen by the eye that views it - the magnification. Kristof Claes summarized it a few posts earlier.

For example, to support a final-image resolution equivalent to 5 lp/mm for a 25 cm viewing distance when the anticipated viewing distance is 50 cm and the anticipated enlargement is 8:

Most importantly, "bokeh" isn't simply "background blur" but all blur outside the DOF; even in the foreground. It's that small lights at a distance are easier to judge bokeh quality.

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Natural materials like copper or aluminum have noteworthy coefficients, while natural stones like granite and marble have more distinct terminal expansion characteristics. That makes these materials essential for use in construction. Polymers like polyethylene's coefficient properties make them a good choice for use in daily products. Composite materials, with their nuanced expansion behaviors, are suited for use in automotive and aerospace applications.

The thermal expansion coefficients of materials are essential metrics that reveal how materials respond to temperature changes. The materials span a range of engineered and natural substances, and their coefficients shed light on their thermal behaviors. The thermal expansion coefficient is represented by α and quantifies the fractional size change of a substrate per unit temperature variation. Positive coefficients indicate expansion and negative ones showcase contraction.

Depth of Field is a function of Focal Length, Effective Aperture, Distance to Subject and Minimum Circle of Confusion. Minimum Circle of Confusion is where things get fuzzy, as that can either be viewed as a function of the imaging medium, or a function of print size.

However, if you really do hold the reproduction ratio constant, the depth of field really is constant. For example, if you have a 20mm lens and a 200 mm lens and take a picture with each at (say) f/4, but take the picture with the 200 mm from 10 times as far away so the subject really is the same size, the two theoretically have the same depth of field. That happens so rarely, however, that it's mostly theoretical.

The above formulas only apply when the distance \$s\$ appreciably is larger than the focal length of the lens. As such, it breaks down for macro photography. When it comes to macro photography, it is much easier to express DoF in terms of focal length, relative aperture, and subject magnification (i.e. 1.0x):

The same is true with sensor size: in theory, if the reproduction ratio is held constant, the sensor size is completely irrelevant. From a practical viewpoint, however sensor size matters for a very simple reason: regardless of the sensor size, we generally want the same framing. That means that as the sensor size increases, we nearly always use large reproduction ratios. For example, a typical head and shoulders shot of a person might cover a height of, say, 50 cm (I'll use metric, to match how sensor sizes are usually quoted). On an 8x10 view camera, that works out to about a 1:2 reproduction ratio, giving very little depth of field. On a full 35mm size sensor, the reproduction ratio works out to about 1:14, giving a lot more depth of field. On a compact camera with, say, an 6.6x8.8 mm sensor, it works out to about 1:57.

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Video: Sensor sizes are expressed in inches notation because at the time of the popularization of digital image sensors they were used to replace video camera tubes. The common 1" circular video camera tubes had a rectangular photo sensitive area about 16 mm diagonal, so a digital sensor with a 16 mm diagonal size was a 1" video tube equivalent. The name of a 1" digital sensor should more accurately be read as "one inch video camera tube equivalent" sensor. Current digital image sensor size descriptors are the video camera tube equivalency size, not the actual size of the sensor. For example, a 1" sensor has a diagonal measurement of 16 mm.

Using the “Zeiss formula”, the circle of confusion is sometimes calculated as d/1730 where d is the diagonal measure of the original image (the camera format). For full-frame 35 mm format (24 mm × 36 mm, 43 mm diagonal) this comes out to be 0.025 mm. A more widely used CoC is d/1500, or 0.029 mm for full-frame 35 mm format, which corresponds to resolving 5 lines per millimeter on a print of 30 cm diagonal. Values of 0.030 mm and 0.033 mm are also common for full-frame 35 mm format. For practical purposes, d/1730, a final-image CoC of 0.2 mm, and d/1500 give very similar results.

DOF is not symmetrical. This means that the area of acceptable focus does not have the same linear distance before and after the focal plane. This is because the light from closer objects converges at a greater distance aft of the image plane than the distance that the light from farther objects converges prior to the image plane.

Recognizing that real lenses do not focus all rays perfectly under even the best conditions, the term circle of least confusion is often used for the smallest blur spot a lens can make (Ray 2002, 89), for example by picking a best focus position that makes a good compromise between the varying effective focal lengths of different lens zones due to spherical or other aberrations.

Aperture changes the rate of CoC growth. Wider apertures increase the rate at which out of focus blur circles grow, therefor DoF is shallower. Narrower apertures reduce the rate at which out of focus blur circles grow, therefor DoF is deeper.

How does aperture affect depth of field? It ultimately boils down to the angles of the rays of light that actually reach the image plane. At a wider aperture, all rays, including those from the outer edge of the lens, reach the image plane. The diaphragm does not block any incoming rays of light, so the maximum angle of light that can reach the sensor is high (more oblique). This allows the maximum CoC to be large, and progression from a focused point of light to maximum CoC is rapid:

It's probably also worth pointing out that this (I believe, anyway) why catadioptric lenses are noted for their lack of depth of field. In a normal lens, even when you're using a large aperture some of the light still enters through the central part of the lens, so a small percentage of the light is focused as if you were shooting at a smaller aperture. With a catadioptric lens, however, you have a central obstruction, which blocks light from entering toward the center, so all of the light enters from the outer parts of the lens. This means all of the light has to be focused at a relatively shallow angle, so as the image goes out of focus, essentially all of it goes out of focus together (or a much higher percentage anyway) instead of having at least a little that's still in focus.

Cte thermalexpansion

The coefficient of thermal expansion (CTE) is defined as the fractional change in length or volume per unit temperature change. It quantifies a material's propensity to contract or expand as temperatures change. CTE for an area is mathematically expressed as:

The circle of confusion is a quirky value here, so we'll discuss that later. A useful average CoC for digital sensors can be assumed at 0.021mm. This formula gives you the hyperfocal distance, which isn't exactly telling you what your depth of field is, rather it tells you the subject distance you should focus at to get maximum depth of field. To calculate the actual Depth of Field, you need an additional calculation. The formula below will provide DoF for moderate to large subject distances, which more specifically means when the distance to subject is larger than the focal length (i.e. non-macro shots):

There is one more factor to consider though: with a shorter lens, objects in the background get smaller a lot "faster" than with a longer lens. For example, consider a person with a fence 20 feet behind them. If you take a picture from 5 feet away with a 50 mm lens, the fence is 5 times as far away as the person, so it looks comparatively small. If you use a 200 mm lens instead, you have to back away 20 feet for the person to be the same size -- but now the fence is only twice as far away instead of 5 times as far away, so it looks comparatively large, making the fence (and degree to which it's blurred) much more apparent in a picture.

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If we used the compact camera at the same 1:2 reproduction ratio as the 8x10, we'd get the same depth of field -- but instead of head and shoulders, we'd be taking a picture of part of one eyeball.

Aperture. Wide aperture lenses give you a shallower depth of field. This is probably the least controversial factor! This is important as some lenses have much larger apertures e.g. 18-55 f/3.5-5.6 vs. 50 f/1.8

\$D_\text{n}\$ = Near limit of DoF \$D_\text{f}\$ = Far limit of DoF \$H\$ = Hyperfocal distance (previous formula) \$s\$ = Subject distance (distance at which the lens is focused, may not actually be "the subject")

However, images seldom are viewed at the “correct” distance; the viewer usually doesn't know the focal length of the taking lens, and the “correct” distance may be uncomfortably short or long. Consequently, criteria based on lens focal length have generally given way to criteria (such as d/1500) related to the camera format.

ΔL represents the change in length, α is the coefficient of linear expansion, L0 refers to the start length and ΔT is the temperature adjustment. The linear thermal expansion property is essential to various engineering applications. These applications often include designing components or structures exposed to temperature variations, like bridges.

As an aside, I think it's worth considering what an incredible stroke of brilliance it was to start measuring the diameters of lenses as a fraction of the focal length. In a single stroke of genius it makes two separate (and seemingly unrelated) issues: exposure and depth of field controllable and predictable. Trying to predict (much less control) exposure or depth of field (not to mention both) before that innovation must have been tremendously difficult by comparison...