Off-Axis Mirror Collimated Beam Contour

[Kimusubi]

When you collimate a point source using an off-axis parabolic mirror (OAP) with a circular shape, the beam area of the collimated light becomes more and more elliptical (x-dir. is smaller than y-dir.) as the off-axis angle is increased. Is there a reason why this happens? And is there a way to quantify this? My best guess is that it has something to do with the magnification because the beam spread is further in the x-dir. than the y-dir, but what does magnification mean when speaking of collimated light (infinite focus distance).

 

[M Quack]

It is a bit like looking at a CD. When you look at the face you see a circle. The more you turn it, the more the shape becomes compressed in one direction until it is a line when you look straight at the edge. (This will also work with good old vinyl records, but not with an iPad :-) )

If the mirror is not too close to the source, then you can approximate it as a flat mirror. In the sagittal (y) direction the projected size of the mirror is its diameter D. In the meridional (x) direction the projected size is D/sin(theta), where theta is the (average) angle of incidence on the mirror. theta=0 when the incidence is normal, i.e. the mirror is behind the source.

This becomes a bit more complicated if the mirror is close to the source, as you have to take into account its curvature and the fact that the angle of incidence changes across the mirror. This could probably be treated as a perturbation.

 

[Kimusubi]

That would make sense to me if the mirror was being rotated. But what I can't conceptualize is why it makes a difference if the light source is moved. For example, if I'm looking directly at the face of the mirror (where the collimated light will be propagating from), as long as the mirror doesn't move, the projected area in this direction will always be the same regardless of the off-axis angle. As long as the point source is located at the parent focal length of the mirror, then it seems to me that the light shape should always be the same regardless of the angle. Is there any way you can clarify this confusion for me?

Also, D/sin(theta) goes to infinity at theta = 0 (light source is directly in front of the mirror) - how does this work then?

Sorry for all the questions. I'm just really confused about this one issue.

 

[M Quack]

A parabola has only one focal point. Only when the light source is in that exact position you get a nicely collimated, parallel beam.

If the source moves away from that point, then you get aberations, i.e. the beam after the mirror is no longer collimated and will diverge. Maybe that is what you are seeing.

I can't quite figure out what your question really is. Can you post a drawing of how things are set up and how they are moving?

The projected size should be D*cos(theta), my mistake. Sorry. The maximum size has to be the diameter of the mirror, when the source is directly in front of the mirror. The minimum projected size is zero, when the source is in the plane of the mirror.

 

[Kimusubi]

I'm referring to comparing the diameter/shape of the collimated beam for two mirrors with the same diameter but different off-axis angles and effective focal length (diff. focal lengths to ensure that the point source is in the focal point of both mirrors). This way both mirrors will theoretically collimate the light perfectly, but will have different beam shapes for the collimated light. For example, the first mirror will have a 15 degree angle and the second will have a 45 degree angle - I'm trying to understand why the 15 degree beam shape will be less elliptical than the 45 degree beam shape simple for being at a smaller angle.

I would draw a diagram but I am absolutely terrible at drawing 3D images.