# Why do certain lenses increase angle of view upon close focus?

I recently picked up a superzoom lens, the Nikon 28-300mm. Although I got it primarily for its versatility, my intuition was that a lens that can do 300mm focal length at 50 cm, as indeed it can, would also offer reasonable magnification for macro shots.

I was shocked to discover that, within a distance of about 5 metres, my 105mm macro lens with 2x teleconverter offers a considerably narrower field of view at 210 mm than my 28-300mm lens does at 300 mm! I found a forum thread on this lens which explains:

Anybody expecting to be able to use it as a macro should carefully check the maximum magnification: 0.32x. Being a IF lens, the Nikkor dramatically increases the angle of view upon closer focus. [...] 0.32x at 50cm roughly calculates to a focal length of 92mm at [minimum focusing distance]... so "dramatically" could have been written even in capital letters.

I'd like to understand better what principles of lens construction and/or physics lead to this counterintuitive behavior. On a pragmatic level: it's clear that I can derive the effective field of view at minimum focusing distance from the maximum magnification listed in the specifications, but how do I go about determining the effective field of view at other distances? For example, how would I determine the field of view of my 28-300mm lens at 300mm and 3 metres? Can these be calculated or must they be determined empirically? If they must be determined empirically, are there people who publicly document this sort of thing?

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The principle of physics behind this behaviour is nothing more than the thin lens formula:

``````1/o + 1/i = 1/f
``````

Where o is the object distance (distance from lens to subject), i is the image distance (distance from lens to sensor), and f is the focal length.

For a very large object distance (approaching infinity) the 1/o term drops to zero, hence:

``````1/i = 1/f
i = f
``````

This means a simple 300mm lens will form a focussed image of a very very far away object a distance of about 300mm behind the lens. This means if it is mounted in tube which places the lens 300mm from the sensor then you will get sharply focussed photographs of objects on the horizon.

What about an object close to the lens at a distance of 600mm?

``````1/600 + 1/i = 1/300
1/i = 1/600
i = 600
``````

The same 300mm lens mounted in a 300mm tube will produces images of objects at this distance which are completely out of focus, however if we lengthen the tube to 600mm our close up object is brought into sharp focus.

We have created a "unit focussing" lens. The problem with such lenses is that they massively increase in physical length when focussing.

To avoid such a huge change in physical length in a close focussing lens like the 28-300mm, designers employ "rear focussing" which works by varying the focal length when focussing close. Going back to the thin lens formula, if a 300mm lens mounted at a fixed distance changes to a 100mm lens, focus changes from infinity to:

``````1/o + 1/300 = 1/100
1/o = 1/150
o = 150
``````

One hundred and fifty millimetres (which is pretty darn close!).

You can in theory use the same formulas to work out the relative focal lengths at different focus distances, but with the caveat that in a complex multi-element lens the distance o corresponds to the object distance from the front principal plane and the distance i corresponds to the image distance from the rear principal plane. The location of these planes depends on the lens design and is not frequently specified by the manufacturer.

Ultimately rear focus makes it relatively easy to drop the minimum focus distance down, which allows manufacturers to slap "macro" on the description and sell more lenses, but as focal lengths are by convention always stated with the lens at infinity focus the customer is in the dark about what's actually happening. All you can really do is treat stated focal length and aperture values as approximate values for moderate focus distances only.

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