Ray Optics: Angular Magnification

[cepheid]

Hi,

I did a homework problem in which I was to show that a refracting telescope, modeled as a sequence of two thin lenses separated by a distance equal to the sum of their focal lengths, is capable of producing angular magnification. In my course we have been solving these questions in geometric optics by modelling a paraxial ray as a vector consisting of its perpendicular distance from the optical axis at a given point, and the angle between it and the optical axis: $(y, \theta)$.

The effect of the optical system on this ray was then ascertained by deriving the ray matrix for the telescope, which was simply a cascade of two thin lens ray matrices and one free space propagation matrix. My solution did indeed show that the telescope produced angular magnification. The homework solutions went on to state that angular magnification is, "the most relevant magnification in the imaging of distant objects." My question is simply, why is this true, why is the magnification of $\theta$ important, more so than magnification of y?

 

[jtbell]

With optical instruments, what usually matters is not how large the image actually is (its linear size), but rather how large it appears to be (its angular size relative to your eye). The angular size depends on both the actual size and the distance from your eye.

Consider for example the image of the moon formed by a telescope. The moon is what, 1600 miles in diameter? If you focus the telescope so that the image is at your "near point" (typically about 25 cm in front of your eye), the image is unlikely to be more than a few feet in diameter. (Of course, you usually can't see the entire moon at once through a telescope. I'm extrapolating from the part that you actually can see.) So the linear magnification is very very small, and the angular magnification better reflects the "usefulness" of the telescope.

 

[cepheid]

Thank you, that makes a lot of sense. The moon is huge (compared to a person), but it is very far away, so it "subtends" (if that is the right word) a very small angle in the sky. It has a small angular diameter (1/2 degree). The telescope increases this angular diameter, but the image size relative to the object size is pretty tiny. If I understand you correctly...