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I am a high school student and have learnt about how curved surfaces reflect and refract (in "ray optics").

We were always told that these surfaces were spherical in shape, meaning they were an arc of a circle.

However in mathematics, I have recently learnt the property that parabolic surfaces converge the light rays coming from infinite distance, exactly at the focus. But this property confused me because in physics we were taught that the ones who converge such light rays at the focus are spherical in shape, not parabolic.

So what exactly is the shape of such mirrors? Are we using some approximation in physics when we say that "spherical" mirrors hold such property? What is that approximation, and what is its range of error?

To satisfy my further curiosity, what about thin "spherical" lenses which converge rays coming from infinity at their focus? Are they really spherical or are they parabolic? What is the range of error (if any) in that case?

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    $\begingroup$ A spherical mirror with an aperture that is much smaller than the radius of the sphere is a pretty good approximation of a parabolic mirror with the same aperture and similar focal length, but the spherical mirror costs much less to make using traditional methods. $\endgroup$ Commented Sep 23, 2021 at 20:19
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    $\begingroup$ Reflecting telescopes generally use paraboloid mirrors, although Newton's first one used a spherical mirror. In traditional telescope mirror making, you first grind a spherical mirror and then you carefully adjust the curvature, as explained on this mirror making site. $\endgroup$
    – PM 2Ring
    Commented Sep 24, 2021 at 3:59
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    $\begingroup$ You may enjoy Making mirrors for giant telescopes, which covers telescopes from the early 20th century through to modern mirrors made using spin-casting. $\endgroup$
    – PM 2Ring
    Commented Sep 24, 2021 at 4:12
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    $\begingroup$ Yes, this is one large misconception that ideal mirrors and lenses are spherical, originating from paraxial approximation whose being approximate is not stressed well enough when teaching. Years ago I was myself very surprised when I saw that a transparent cylindrical dish of water didn't focus parallel rays at a single point; only then did I find out that all the school-level theory is just an approximation, rather than result of exact ray tracing. $\endgroup$
    – Ruslan
    Commented Sep 24, 2021 at 15:08
  • $\begingroup$ Useful search term (wrt. lenses) : spherical aberration. $\endgroup$ Commented Sep 25, 2021 at 20:26

4 Answers 4

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Well, the mirrors you are learning in physics are spherical. There are both spherical and parabolic mirrors. The only difference between them is that parabolic mirrors are more precise; they have only one focal point. Spherical mirrors also have one focal point only when the rays coming are paraxial (rays very close to principal axis). When rays hit the mirror far from principal axis they create different focal point creating multiple focal points, collectively known as focal volume.

See the images below:

Enter image description here enter image description here

You can see multiple focal points in concave one, whereas a single focal point in the parabolic one. This is called spherical aberration.

Now, the question arises: if parabolic mirrors are more efficient than spherical mirrors, why even make spherical ones?

For optical applications, like Newtonian telescopes, the illustrations here are greatly exaggerated. Telescope mirrors are much less curved, almost flat. And parabolic telescope mirrors look spherical and very nearly are spherical, deviating from the sphere by perhaps only millionths of an inch.

In reality, all optics suffer from diffraction. 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. If a spherical mirror is a small enough section of a sphere of large enough radius, then it can still be diffraction limited. Small Newtonian telescopes, commonly around 114 mm diameter and 900 mm focal length, usually have spherical mirrors and are diffraction limited or nearly so. Other kinds of telescopes use spherical mirrors, but correct the spherical aberration with lenses or other optical elements.

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    $\begingroup$ @J.Murray because semi spheres are easier to manufacture. Have been for so long that the discrepancy between the two has a term: spherical aberration. $\endgroup$ Commented Sep 23, 2021 at 20:06
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    $\begingroup$ @candied_orange besides being difficult to manufacture the parabolic mirror has the worst coma aberration, a compensated spherical mirror has a much larger field of view, e.g. en.wikipedia.org/wiki/Arecibo_Telescope $\endgroup$
    – hyportnex
    Commented Sep 23, 2021 at 20:19
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    $\begingroup$ @TheSpaceGuy I would suggest editing your answer. The general rule on PhysicsSE is that comments should always be treated as temporary and subject to deletion once they've served their (limited) purpose, and if there's information you want to convey you should edit it into your answer. You need not worry about stepping on commenter's toes, either - if a user wants their name attached to some piece of information, then they should write their own answer. $\endgroup$
    – J. Murray
    Commented Sep 24, 2021 at 3:12
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    $\begingroup$ The answer is broadly right but makes unwarranted claims at the end. In practice glass lenses are not usually very curved and the comments that parabolic mirrors have a"narrow opening" and "too curvy, therefore fragile" are just nonsense for real world mirrors. $\endgroup$
    – matt_black
    Commented Sep 24, 2021 at 14:56
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    $\begingroup$ I like to think that all of them; spheres, ellipsoids, paraboloids, hyperboloids) have two mathematical focal points, but paraboloids have one at infinity, hyperboloids have one behind and spheres have two (if you define an optical axis) on top of each other. But that's just me. +1 $\endgroup$
    – uhoh
    Commented Sep 25, 2021 at 10:16
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Either there was some qualification that you missed, or your physics class is being overly simplistic. I don't know whether you've taken Calculus yet, but in Calculus terms, spheres and parabolas are second order approximations to each other. 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). The smaller the width of the mirror, compared to the radius of the sphere, the less aberration there is.

The Taylor series for a circle is $1-\frac{x^2}2-\frac{x^4}8-\frac{x^6}{16}...$, so if compared to the parabola $1-\frac{x^2}2$, the error will be on the order of $\frac{x^4}8$, and the derivative (which determines the angle of reflection) will be on the order of $\frac{x^3}{2}$. So if you have a mirror with width one tenth the radius, the error in the slope will be about one part in $2000$.

There is also spherical aberration in lenses (also, for lenses, parabolas are not the shape to eliminate aberration).

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.

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Additional answer:

As much as "nature abhors a vacuum" it also abhors glass surfaces that aren't spherical. 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.

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.

But then, instead of just a focal length or radius of curvature, you've got to specify that aspherical term carefully. 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.

Most complicated lens systems for cameras or projection systems of all types are made out of a collection of lenses with all spherical surfaces.

They get the aberrations corrected by adding a few more spherical surfaces rather than adding one aspherical surface.

From 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. The aspherical element may only have one side aspheric.

Nikon AF-S 35mm f/1.4G

Nikon AF-S 35mm f/1.4G

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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:

  • If the incident ray goes through the center of curvature, the reflected ray also goes through the center of curvature.
  • 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.

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.

Fortunately, when curvature is small, a parabola and a circle are very similar. Therefore, real concave mirrors can be built spherical or parabolic (or somewhere in-between) with small curvature and divergences from an ideal mirror will be small enough to be accounted and corrected as aberrations, as the accepted answer says.

The reason for circles and parabolas being similar is explained by Accumulation's answer but since it involves calculus, which in high school is taught several years after geometrical optics, high school students are left with ideal mirrors without being told that they are just ideal - although useful.

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