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Many lenses (like the Pentax DA 15mm f/4 Limited, or the Nikon AF-S 35mm f/1.4G) are described as having "aspherical lens elements". Does this mean that regular lenses are spherical lenses? What's the difference, and what are the advantages of one over the other?

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A lens marketed as "aspherical" will usually have only one surface (one side of one piece of glass) aspheric, and all the other surfaces will be spherical (or flat).

So the vast majority of glass in any lens, even those lenses marketed as aspherical, is spherical anyway.

An aspheric surface can help correct for spherical aberration, which can make the image look soft when the aperture is wide open. But it's not an exact science, because compromises have to be made. For example, the spherical aberration correction can sometimes make background bokeh more harsh, which isn't great for portraits.

Aspheric lens surfaces are much more expensive to produce, because they can't be ground by a natural rotary motion.

Fun fact: high quality aspheric lens surfaces have been around at least since 1667, used in telescopes, reading glasses and burning glasses (!).

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You mean since the late tenth century: Ibn Sahl is credited in that same Wikipedia article with "work[ing] out the shapes of anaclastic lenses that focus light with no geometric aberrations"; in other words, with designing aspherical lenses. In a 1990 Isis article, Roshdi Rashed documents Ibn Sahl's geometric construction of refracting optical instruments based on conic sections, such as the hyperbola, with perfect--aspheric--focusing properties. He points out that an 11th century successor, Ibn al-Haytham, further addressed astigmatism and aberration. – whuber Apr 13 '11 at 1:50
Erm, why can't they be ground by a natural rotary motion? Aspheric lens elements are still symmetric across the center axis; so any technique that works for spheres should work for paraboloids as well. – Billy ONeal May 30 '12 at 19:08
It's still more complicated to grind something a parabolic shape than spherical - even though the lens can be spun around on its centre axis. – thomasrutter May 31 '12 at 1:39
+1 just for the fun fact alone. – Sachin Kainth Aug 29 '13 at 11:06
But it's not an exact science It is exact science of optics, however each manufacturer makes his own choice for f-number, contrast, weight etc – aaaaaa Jun 16 '15 at 6:59

From what I have read and and seen about aspherical lenses, they are designed to prevent aberration of light hitting the surface of the lens. This means, it prevents image distortion for the person wearing the lenses. This is accomplished by making the edges of the lens curve out, away from the curve of the spherical design of the lens, making it a sphere in the middle, and tapered at the edges.

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It is simple and relatively cheap to make lenses whose surfaces are parts of a plane or parts of a sphere. Such lenses do not focus light perfectly; this lack of focus is spherical aberration. This graphic from a Wikipedia article schematically illustrates how the light rays fail to converge (lower half) compared to a hypothetical perfect lens (upper half).

enter image description here

Spherical aberration is especially prominent in large, fast (bright) lenses. This lack of focus shows up as various forms of fuzziness. The problem can be corrected by placing other lenses in front of and behind the lens. It can also be corrected by changing the shape of the lens surfaces (making them aspherical), but that tends to be more difficult and expensive to carry out: glass spherical shapes are simple to make and measure; aspherical ones are not.

When additional lens elements are added as correction, they increase the amount of flare and reduce the contrast of the photograph, as well as adding to the size and weight of the lens. They might also alter the colors a little. Therefore, aspherical elements typically signal a lens that is crafted to produce contrasty, colorful images and to be lighter and easier to use. They do not in themselves assure high quality, because lenses can exhibit other problems besides spherical aberration. For instance, small aspherical lenses can be molded of plastic and routinely appear in cheap photo equipment. However, a large glass aspherical lens is much more expensive to produce and usually reserved for better lenses.

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The photographer's answer is that it does not matter. We let optical engineers find the best way to build lenses and then use them for our craft.

The technical answer is yes, non-aspherical lenses are spherical in the sense that one of their surfaces corresponds to the outer surface of a sphere of some radius.

Aspherical lenses are more complex and are not constrained to following the curve of a sphere. That also means there are lots more variance among aspherical lenses. This gives optical engineers more freedom when designing such lenses and therefore more chances to correct for optical issues.

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