1. Depth of focus. For a given object distance , the depth of focus is the range of image distances for which the object is in focus.

It is to be noted, however, that the thin lens equation suggests that every object distance is imaged to a different image distance. In other words, the image of an object which is 15 feet away will appear at a different location behind the lens than the image of an object which is 10 feet away. If a camera (or an eye) is to be useful, it must be able to image objects at different distances. In a camera, one can adjust focus by changing the distance between the lens and the film; in the human eye, muscles in the eye change the focal length of the lens by distorting its shape.

Magnification formulaforlensin terms of focal length

…and a whole lot of light! Something that is implied but not stated in this article is that the more you reduce the diameter of the entrance pupil of the camera, the more less light reaches the focal plane. To take a picture through a pinhole, you have to either increase the illumination of the scene or use a longer exposure.

A lens is a device made of a material with a different refraction index to air (there can even be electromagnetic lenses that act on electric currents). This and its shape allows it to bend rays of light as they come into contact with it.

In the case of violent kangaroos, it may be better to go for the second option: that's why camera manufacturers sell extension tubes, short rings to mount between the lens and the body, which end up increasing hhh by some precious millimeters.

Lenses can focus or "unfocus" light rays. In this tool, we will only consider converging lenses. Their main feature is the ability to focus every ray entering the lens parallel to the optical axis at a specific point, the focus.

While stumbling through the ‘tubes the other day, I came across a very cute series of photos, an example of which is pictured below:

If you'd like to know more, try our others optic calculators dedicated to lenses, like the thin lens equation calculator or the lens maker equation calculator.

Clearly I hadn’t! (This confirms my often-told joke about my work in ‘theoretical optics’: “As soon as it can be applied, I change topics.”) The tilt-shift lens is quite cool, though, and it’s now clear how it works.

Our lens magnification calculator will focus on the world of lenses in photography, finally explaining what magnification is, why it is different from zoom, and much more!

For this example, the object is placed at , which by the thin lens equation results in an image at . For the picture, the -number is , though it is important to note that the -number has no effect on the axial position of the image. It will, however, play an important role in the discussion of depth of field and depth of focus.

🔎 The word "focus" comes from Latin for "fireplace". This is because the Romans believed that their ancestral gods were located in the fireplace, or hearth, and so would direct (or focus) their worship towards it.

2. Depth of field. For a given image distance , the depth of field is the range of object distances for which the object is in focus.

The values of hhh and ggg are hidden in the further magnification properties section of our calculator, so if you need to know either of these, just click the button!

🔎 Lenses and their properties have been known by humanity for a long time. However, only in the 13ᵗʰ century did lens-making skills reach a level of refinement that allowed for the construction of glasses, telescopes, and much more!

This effect is a clever exploitation of the concept of depth of field, and its related ‘cousin’, depth of focus. I thought I’d do a short post about the idea (my friend Personal Demon suggested the topic months ago, actually) and the related mathematics. Along the way we’ll introduce a few simple lens topics from geometrical optics.

I’m not a digital photo expert, but it looks like a number of ‘tricks’ were used to make the pictures look miniature, including the simple psychological effect of always having views from above. The most significant modification, though, is of the focus: in the modified pictures, the foregrounds and backgrounds are out of focus, and this alone can fool the eye. Why is this?

Power oflens formula

The magnification of a lens is an absolute value that depends on the focal length of the lens itself, while the zoom is a relative quantity that describes how much you can change the focal length of a lens by, thus changing its magnification.

Though you probably don’t consciously realize it, your brain is taking some of its distance cues from this depth of focus behavior. The ‘miniature’ pictures help make big objects appear small by simulating a narrow depth of field.

Can we ‘fake it’ the other way, and make miniature objects appear larger? In principle this could be done by taking pictures of miniatures with a pinhole camera. I’ve talked about the image-forming abilities of the camera obscura in a previous post; a figure of this is shown below:

Thanks to the properties of similar triangles, we can compute the magnification of a lens also using the distances between the object/image and the lens:

Neither of these concepts are ‘absolute’ properties of an imaging system: they both depend on the resolution requirements at the end of the system. For instance, a high-resolution digital camera requires its images to be in much sharper focus than my 20/200 near-sighted eyes do. A point object which is being improperly imaged will produce a circular spot, rather than a point image, as illustrated below. It can be seen that the object on axis at A will produce a point image at the point I; the object on axis at B, however, will produce a point image at an earlier location. At I, object B produces a circular spot. It is important to note, and one can see from the diagram below, that the size of the circular spot depends upon the size of the lens aperture (-number).

Dr. Skull wrote: “Of course, if you try to use a pinhole camera to make a miniature scenery look real, you had better have some extremely well-detailed miniatures!”

PD wrote: “To take a picture through a pinhole, you have to either increase the illumination of the scene or use a longer exposure.”

stuwat wrote: “In fact, it could probably be understood even without the mathematics, but it makes for a more comprehensive explanation for those who are willing to follow it.”

For our purposes we only need two numbers to characterize the properties of a lens: its focal length and its -number. The focal length is the distance between the center of the (presumably thin) lens and its focal point, as illustrated above. The -number is the diameter of the usable part of the lens, usually expressed in units of focal lengths. In other words, a lens whose diameter is half of a focal length has an -number of , while a lens whose diameter is an eighth of a focal length has an -number of .

The perceived magnification of an object, thanks to the use of powerful telephoto lenses, comes from the reduced projection of the object onto relatively small sensors. If that projected image can change the size, let's say by a factor of two, we say that the lens has a 2× zoom.

This distance is known as the hyperfocal distance. When a lens is focused at or beyond the hyperfocal distance, everything beyond that distance is always in focus. Herein lies the key to understanding the ‘miniature’ pictures mentioned at the beginning of this post. When we focus our eyes on an object that is extremely close to us, we have a relatively narrow depth of field: objects immediately in front of and behind the object we’re looking at are out of focus. When we focus our eyes on a distant subject, we are looking past the hyperfocal distance and see almost everything in focus. Incidentally, one of the things that happens when you turn a digital camera onto ‘landscape’ mode is that the focus is set at or beyond the hyperfocal distance.

The magnification of a lens is an absolute measure of how much the height of a real image differs from the object's height. Remember, that in a camera, the real image forms on the sensor (or on the film, if you're old school).

Quite right. Even looking at a single aspect of photography such as depth of field, one realizes how many factors must be taken into account to make a well-crafted image. Aperture size, focal length, lighting, lens type and other factors all contribute, and are not necessarily independent of one another. This complexity explains why most of my photographs end up looking like a Yeti riding the Loch Ness Monster while being illuminated by a UFO.

Their lenses are usually manufactured with a focal length of 25 cm25\ \text{cm}25 cm. If you use the lens to look at an object closer to it than that distance, you create a virtual image of the object.

clearly you have not heard of the wonderful invention that is a tilt-shift lens. although this can be done in photoshop, the lenses do a better job.

Magnification formulafor mirror

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The magnification of a lens with focal length 55 mm at a distance of 100 m is m = 0.0005506. To calculate it, follow the steps:

Since that beast would be too dangerous to photograph at a close distance, we suggest you use a 500 mm500\ \text{mm}500 mm telephoto lens. We advise you to keep your distance, let's say 150 meters (but remember that a kangaroo can reach a maximum speed of 70 m/s70\ \text{m/s}70 m/s). Insert these values into our magnification of a lens calculator, which will return:

Of course, if you try to use a pinhole camera to make a miniature scenery look real, you had better have some extremely well-detailed miniatures!

Magnificationoflens formulaClass 10

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Physicist by education, scientist by vocation. He holds a master’s degree in complex systems physics with a specialization in quantum technologies. If he is not reading something, he is outdoors trying to enjoy every bit of nature around him. He uses his memory as an advantage, and everything you will read from him contains at least one “I read about this five years ago” moment! See full profile

Now consider that the sensor is at most a few centimeters wide, while you can take a picture of the Eiffel Tower, which is 330 m330\ \text{m}330 m tall. Even from afar and with a powerful telephoto lens, you'll always get a magnification that is much smaller than you expect when taking pictures with a camera.

The lens is taken to have an aperture , and we assume for future sanity that all objects lie at distances greater than the focal length . We choose an image plane which lies at distance ; images which focus at distances (red lines) and (blue lines) both produce a circular spot at the image plane ; it is assumed that this spot is of diameter , equal to the circle of confusion. A little bit of geometry gives us the following formulas for and ,

Magnification formulaoflensin terms of v and u

The zoom describes how much the lens's focal length can change by (there are such things as zoom lenses). A typical 18/5518/5518/55 lens will have its zoom defined by:

In photography, the magnification of a lens is the ratio between the height of the image projected onto the sensor or film of the camera and the height of the real image you are taking a picture of.

As you can see, now the rays on the right side of the lens do not converge. We are dealing in terms of virtual images, which originate from the virtual continuations of the rays, creating a non reversed image of the object.

Magnification formulafor convexlens

A lens can be used to focus light through the process of refraction. Light rays passing through the glass of the lens are deflected at different angles depending on the angle at which the ray approaches and the thickness of the glass at the point of entry. For instance, the image below shows the effect of a lens upon a collection of parallel rays:

Imagine you are taking a picture of a huge kangaroo, let's say two meters tall and weighing 95 kg95\ \text{kg}95 kg, like the one that terrorized Brisbane a few years ago.

That’s a pretty cool effect. Some of the photo’s come out looking better than others, the better ones appearing so much like miniatures that you really have to take a closer look. The explanation was also really well done. In fact, it could probably be understood even without the mathematics, but it makes for a more comprehensive explanation for those who are willing to follow it.

I think he explained it the best way to understand, I liked when the writer said I am not a digital photo expert. So, you did the best.

It is to be noted that lies closer to the lens than , because lies closer to the lens than . We can transform our equations for the image locations into their corresponding object locations using the thin lens formula. After a bit of work, one finds that

We may also ask what happens to the images of objects which are not located at the ideal object distance . Again, there will be a range of distances around the ideal distance where the blur will not be noticeable. This range of distances is known as the depth of field.

What can we learn from these formulas? The formulas for and are identical except for a minus sign which appears in the denominator of . This sign makes all the difference, however, because that denominator can therefore go to zero, which means that becomes infinite. This happens when the lens is focused onto an object at a distance such that

For me, the most obvious ones are the ones which show too much, e.g. the first one. When you can see for miles into the distance, it’s hard to imagine a miniature of that size being constructed.

where is the distance of the object to be imaged from the front of the lens and is the distance at which the image appears behind the lens. This is illustrated below:

What happens if our film (or retina) is not placed at the ideal image distance ? The image will be increasingly blurry the further we move from the ideal image distance. There will be, however, a range of distances around the ideal distance where the blur will not be noticeable. This range of distances is known as the depth of focus.

Tom wrote: “I think the ones with people are more recognizable as real because it’s difficult to get miniatures with realistic body proportions and poses.”

Cool. I’ve seen pictures like this before; I think the ones with people are more recognizable as real because it’s difficult to get miniatures with realistic body proportions and poses.

Is it a picture of a real place, or a picture of a miniature model? All the pictures are apparently of real places, digitally modified via Photoshop to make them look tiny! If you are skeptical, you can see the original picture of Stonehenge in the snow here, and what looks to be the original of Tokyo Tower here.

The ideas of depth of focus and depth of field both concern the ability of an image-forming device to make sharp images, but under different circumstances. When we are focusing on a single object and want to know the range of positions within the camera which are effectively in focus, we are referring to depth of focus. When we are imaging a collection of objects at different distances and want to know how many of these objects will be in focus at a given image plane, we are referring to depth of field.

Magnification formulaBiology

The image of a camera obscura (or, with film, the pinhole camera) is created by simple geometric projection. A perfect image is formed regardless of object or image location relative to the pinhole, which means that the pinhole camera has infinite depth of focus.

Now we can do some math to quantify these concepts! We’ll restrict ourselves to studying depth of field, as it is directly relevant to the ‘miniature’ pictures listed earlier. Let’s start by looking at the behavior on the imaging side of the lens, as shown below:

Expand the further magnification properties section to see the variable extension tube. We set it at 0 mm0\ \text{mm}0 mm by default, but change it according to your needs!

Lens formula magnificationpdf

Since, in most cases (unless you are using a microscope), the lens shrinks the object, the magnification value is less than 1.

The thin lens model can be used to understand the operation of simple cameras and the human eye. In a camera, a lens (or, usually, a system of lenses) projects an image onto photographic film or a CCD sensor. In the human eye, a lens projects an image onto the retina in the back of the eye, where the image information is converted into neuron signals:

Update: The original website with its digitally-altered ‘miniworld’ pics seems to have been taken down.  I’ve constructed a few of my own ‘miniworld’ pics to give the reader an idea of what I’m talking about:

A camera is nothing but lenses and a sensor. At least in theory! To understand how it works, we need to explore the world of optics.

However, when talking cameras, the magnification is usually a really small number. The number followed by a ×\times× is the zoom.

Maybe you expected the magnification to be a bigger number, something like 10×10\times10× or 20×20\times20×, like the values you see on binoculars or telescopes (we made an entire calculator for that, check out our telescope magnification calculator).

First thing – the upward facing arrow on the left of the image is the object we are looking at. The rays of light coming from it hit the lens. The one parallel to the optic axis (the topmost line) gets focused and so converges on the focus. The ray passing through the center of the lens meets the focused ray on the other side of the lens, which creates a flipped image called the real image of the object.

We define the circle of confusion as the largest diameter circular spot we are willing to tolerate in our imaging system.

When you are snapping a picture, you don't usually know the values of hhh and ggg, but you know the focal length for sure, and you likely know the distance between you and your subject. These two quantities are enough for you to calculate the magnification of your lens!

Without going into too much detail, according to the theory of geometrical optics, in free space light travels along straight line paths called rays. One can see how this idea came about: light passing through gaps in a cloud or through gaps in the foliage of trees appears as a continuous line of light.