Ray Optics Doubt: Can an Equilateral Triangle Form?

[UchihaClan13]

Alright I was doing some basic questions on ray optics when this doubt came to my mind
So my doubt goes like this:(IT may be silly)
Consider a concave mirror with pole P and centre of curvature C
Its principal axis is extended on both sides
Now consider a ray parallel to the principal axis
Call this ray BA (it strikes the concave mirror at A)
Now assume it makes an angle 60 degrees with the normal
Thus as per the laws of reflection,angle BAC=angle CAF(a ray parallel to the principal axis passes through the principal focus after reflection
Then angle ACF=60 degrees(as BA is parallel to the principal axis thus due to alternate interior angles they are equal)
Which results in angle CFA=60 degrees(as CAF is a triangle)
As all angles are equal ,CAF should be an equilateral triangle
However CA=2CF(since CA is the radii of curavture and CF=FP which is the focal length)
This violates the condition for an equilateral triangle
I tried to solve this paradox but am still getting confused
Some help or insight would be much appreciated
Thanks

 

[Merlin3189]

Yes, you have spotted the flaw in spherical mirrors! Spherical aberration.
For a spherical concave mirror a ray from C goes back to C and a ray towards the pole reflects at an equal angle on the other side, but the parallel ray does not pass through the focus. So a spherical mirror is only an approximation to the ideal mirror of school optics.
Provided the mirror is small enough in relation to the radius of curvature (or the radius of curvature is big enough in relation to the aperture) then the parallel ray goes near enough through the focus at half the radius of curvature. The further away from the axis you get, the shorter is the point where the parallel rays focus..

If the mirror is parabolic instead of spherical, then the parallel ray does indeed pass through the focus (but then there is no centre of curvature, so you use the parallel rays and possibly the polar symmetric ray.)

Spherical mirrors were (and are) used because they are easier to make. Parabolic mirrors are also not perfect, they have other aberrations.

 

[UchihaClan13]

Oh I see
But then why is spherical aberration not considered in various places
And why are ray diagrams full of flaws
You did clear my doubt but could you be a bit more specific taking my case as the example
Thanks!

 

[davenn]

Spherical mirrors were (and are) used because they are easier to make. Parabolic mirrors are also not perfect, they have other aberrations.

indeed ... even fast, f5 or less ( f4 etc) parabolic mirrors also tend to suffer spherical aberration as well as chromatic aberrationDave

 

[Drakkith]

indeed ... even fast, f5 or less ( f4 etc) parabolic mirrors also tend to suffer spherical aberration as well as chromatic aberration

How so? A paraboloidal mirror should not have any spherical aberration (at least paraxially, not sure about off-axis) nor should it suffer from chromatic aberration.

 

[davenn]

both my fast, f4.5 Newtonians suffered from both, chromatic aberration being the worse around the outer edges of the FOV

EDIT: ... well reading tells me otherwise for a parabolic mirror, and assuming my mirrors are parabolic and not spherical ... then what I have been told in the past as the reason for the aberrations ( both)
would be incorrect. This leaves me now wondering the real cause of these aberrations ?

Dave

 

[jtbell]

Spherical mirrors were (and are) used because they are easier to make.

And we discuss them, along with spherical refracting surfaces for lenses, in introductory optics because they're easier to analyze. You get nice simple equations like the ubiquitous thin-lens equation 1/p + 1/q = 1/f.

 

[Andy Resnick]

both my fast, f4.5 Newtonians suffered from both, chromatic aberration being the worse around the outer edges of the FOV

EDIT: ... well reading tells me otherwise for a parabolic mirror, and assuming my mirrors are parabolic and not spherical ... then what I have been told in the past as the reason for the aberrations ( both)
would be incorrect. This leaves me now wondering the real cause of these aberrations ?

Dave

Perhaps is just confusion between 'off axis' and 'image height'- parabolic mirrors have large positive coma. That is, off-axis object points are imaged with varying magnification. It's also stigmatic for infinite object distance only.

 

[Merlin3189]

Now consider a ray parallel to the principal axis
Call this ray BA (it strikes the concave mirror at A)
Now assume it makes an angle 60 degrees with the normal

So the diameter of the mirror must be at least R √3 or about 3.5 x the focal length, well beyond the size where the mirror formula is even approximately correct.

My own cheap telescope has an aperture of about f/8 (1/8 of the focal length) and I'd have thought that is about as big as you can reasonably go.with a simple Newtonian.

Your geometric point, is that CFA is an isosceles triangle. This can only be true when A is on the axis and CFA is a straight line. Then FA = f = PC/2 = R/2.
But as you move away from the axis,FA increases in length.
If I get time, I might have a look at working out an expression, but I don't think it will be a simple one. The mirror assumption is correct only when A is right on CFP and as A moves away the focus gets nearer the mirror. Also away from the axis rays are converging/diverging more steeply than near axis rays, so the effect of changes in focal length on the circle of confusion is also greater. (Sorry I can't explain that better without a diagram.) In short, the further you are from the axis, the worse everything gets.

And we discuss them, along with spherical refracting surfaces for lenses, in introductory optics because they're easier to analyze. You get nice simple equations like the ubiquitous thin-lens equation 1/p + 1/q = 1/f.

Fair enough, maybe.
Presumably you get the same formula (1/p + 1/q = 1/f) for a parabolic mirror?
As for the analysis, I guess maths is easier with a circle, but one definition of a parabola is the locus of points equidistant from a point and a line, or even for a paraboloid equidistant from a point and a plane. Then using the equal time principle, that point must be the focus for all paraxial rays(*). The axis of symmetry is perpendicular to the tangent (plane or line) where it intersects the parabol(a/oid) so the ray through the pole behaves as expected. And there is no centre of curvature, so you do your ray diagrams with the other two. (I presume with spherical mirrors we should really use only the rays through C and P, but F is so convenient, isn't it.)

DaveN presumably you have a lens eyepiece. Maybe that's the source of your chromatic ab? If the image from the mirror isn't all where it should be, then the lens corrections optimised for the ideal image might produce aberrations with the rest? And are you sure it's parabolic? But I don't know much about telescopes, so that's just speculation.

(*) or maybe I should say for a plane wave perpendicular to the axis. Tho, as far as I can see, it all boils down to the same thing.

 

[davenn]

Perhaps is just confusion between 'off axis' and 'image height'- parabolic mirrors have large positive coma. That is, off-axis object points are imaged with varying magnification. It's also stigmatic for infinite object distance only.

Uh huh you may well be right, and all those years ago I was taught incorrectly ...
Ahh well, nothing like learning the new and correct ways

thanks to you and Drakkith :)

Dave

 

[Drakkith]

EDIT: ... well reading tells me otherwise for a parabolic mirror, and assuming my mirrors are parabolic and not spherical ... then what I have been told in the past as the reason for the aberrations ( both)
would be incorrect. This leaves me now wondering the real cause of these aberrations ?

The overwhelmingly dominant aberration in a fast paraboloidal (parabolic) mirror is coma. Eyepieces themselves, or other lenses you may have in the optical path, can introduce other aberrations, but unless these lenses are of extremely poor quality, coma is still the dominant aberration.

 

[UchihaClan13]

Understood it completely
I have another point to supplement your explanation(Merlin3189)
That is that for such a ray which makes a big angle like 60 degrees,paraxial approximation is out of the question
And when you use the sine rule taking some arbitrary point O instead of the focus(since spherical aberration occurs)
and take PO as x and OC as r-x
You get x=r-r/2sectheta and another value at the same time
Which means x equals 0 and some other value a at the same time
Which is impossible as you said,the diameter or radius has to be quite large,as compared to the focal length
Hence no equilateral triangle is formed here
Am I Right?
Thanks a lot guys for your help !:)