The problem is that the first rule is true only for spherical mirrors, while the second one is only true for parabolic mirrors, which at first sight makes high school geometrical optics disappointingly wrong.įortunately, when curvature is small, a parabola and a circle are very similar. If the incident ray is parallel to the axis, the reflected ray goes through the focus, and vice-versa - or alternatively, if several incident rays are parallel, their reflected rays intersect at the focus.If the incident ray goes through the center of curvature, the reflected ray also goes through the center of curvature.In geometrical optics as taught in high school (at least as it was taught in mine), treatment of concave mirrors is based on two rules: The aspherical element may only have one side aspheric. They get the aberrations corrected by adding a few more spherical surfaces rather than adding one aspherical surface.įrom Photography SE's How does a spherical lens differ from an aspherical lens? includes the example of a Nikon AF-S 35mm f/1.4G with the following image, showing that they chose to add one aspherical element to this expensive compound lens with 10 glass lenses assembled as 7 elements and a total of 17 different surface shapes. Most complicated lens systems for cameras or projection systems of all types are made out of a collection of lenses with all spherical surfaces. While a reflecting telescope mirror might want to be a parabola, aspherical lens surfaces suddenly become very specific and therefore more expensive being single purpose. So unless you really need it, it's easier to get by with a sphere and spherical surfaces are still the norm unless you go and specify an asphere.īut then, instead of just a focal length or radius of curvature, you've got to specify that aspherical term carefully. The process of grinding lens or mirror surfaces produces spherical surfaces because those are the only ones that can be slid around over each other perfectly. In general, if you are told different things in math and physics class, it's probably safe to assume that the physics class is taking an approximation and/or special case.Īs much as "nature abhors a vacuum" it also abhors glass surfaces that aren't spherical. There is also spherical aberration in lenses (also, for lenses, parabolas are not the shape to eliminate aberration). So if you have a mirror with width one tenth the radius, the error in the slope will be about one part in $2000$. The Taylor series for a circle is $1-\frac$. The smaller the width of the mirror, compared to the radius of the sphere, the less aberration there is. That is, you can have a sphere and a parabola that have the same first and second derivative, and they will differ only in third order (in fact, since they're both even function, they will differ in the fourth order terms). I don't know whether you've taken Calculus yet, but in Calculus terms, spheres and parabolas are second order approximations to each other. Other kinds of telescopes use spherical mirrors, but correct the spherical aberration with lenses or other optical elements.Įither there was some qualification that you missed, or your physics class is being overly simplistic. Small Newtonian telescopes, commonly around 114 mm diameter and 900 mm focal length, usually have spherical mirrors and are diffraction limited or nearly so. If a spherical mirror is a small enough section of a sphere of large enough radius, then it can still be diffraction limited. If the spherical aberration causes less image degradation than diffraction, then little or nothing is gained by using a parabola, which is harder to make. In reality, all optics suffer from diffraction. And parabolic telescope mirrors look spherical and very nearly are spherical, deviating from the sphere by perhaps only millionths of an inch. Telescope mirrors are much less curved, almost flat. Now, the question arises: if parabolic mirrors are more efficient than spherical mirrors, why even make spherical ones?įor optical applications, like Newtonian telescopes, the illustrations here are greatly exaggerated. You can see multiple focal points in concave one, whereas a single focal point in the parabolic one. When rays hit the mirror far from principal axis they create different focal point creating multiple focal points, collectively known as focal volume. Spherical mirrors also have one focal point only when the rays coming are paraxial (rays very close to principal axis). The only difference between them is that parabolic mirrors are more precise they have only one focal point. There are both spherical and parabolic mirrors. Well, the mirrors you are learning in physics are spherical.
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