The biggest possible size of an object on the eye without the aid of an optical instrument is obtained by placing it at the near point of the eye $N$. If we bring an object closer than the near point then it will occupy a larger angle, but we won't be able to focus on it. This problem can be addressed by a magnifying glass. If we place an object closer than the focal length of the lens, it will produce a virtual image at a distance $d_{i}$. The maximum magnification is achieved by bringing the lens right up to your eye and then arranging the lens, object and your head so that the image is at the near point. To find the magnification we need to know $d_{o}$ which can be found from the lens equation, taking $d_{i}=-N$

$\frac{1}{d_{o}}=\frac{1}{f}-\frac{1}{d_{i}}=\frac{1}{f}+\frac{1}{N}$

In the small angle approximation

$\theta=\frac{h}{N}$ and $\theta'=\frac{h}{d_{o}}$

The angular magnification of the lens, also called the magnifying power is defined as $M=\frac{\theta'}{\theta}$ which we can see here is

$M=\frac{N}{d_{o}}=N(\frac{1}{f}+\frac{1}{N})=\frac{N}{f}+1$

It is not very convenient to use a magnifying glass with the eye focused at the near point, firstly as we are required to constantly maintain the correct positioning of lens, object and head, but also because our eye muscles are at maximum exertion, which is not very comfortable over long periods of time. An alternative way to use a magnifying glass is to place the object at the focal point of the lens producing an image at $\infty$.

In this case

$\theta'=\frac{h}{f}$

and the magnifying power is

$M=\frac{\theta'}{\theta}=\frac{N}{f}$

A magnifying glass is only useful for looking at objects nearby (the maximum object distance from the lens is the focal length). To view distant objects we need to use a telescope. A refracting telescope uses two lens, and objective and an eyepiece. The objective produces an image which is then magnified by the eyepiece.

The original apparent object size is

$\theta\approx\frac{h}{f_{o}}$

If we consider an object at infinity and an eyepiece which is adjusted so that the focus of the eyepiece is at the focus of the objective (this produces a final image at infinity, which is why I chose not to draw it this way on the diagram), then the apparent size of the final image is

$\theta'\approx\frac{h}{f_{e}}$

giving the magnification power of the telescope as

$M=\frac{\theta'}{\theta}=-\frac{f_{o}}{f_{e}}$

with the minus sign signifying that the image is inverted.

Several fabrication difficulties involved in making large refracting telescopes can be overcome by making reflecting telescopes which use mirrors in the place of an objective lens. It is much easier to make a large mirror than a large lens, and also make it parabolic so it does not suffer from spherical abberation. As they work on reflection, instead of refraction mirrors do not suffer from chromatic abberation.

Spherical aberrations are present for mirrors and lenses that have a spherical surface. Parabolic surfaces are much harder to produce so are not as widespread

Astigmatic aberrations are due to azimuthal asymmetric defects or “out of roundness”

Chromatic aberrations are present for lenses that have an index of refraction that varies with wavelength. Mirrors typically do not have this problem.

Magnetic systems used for spectroscopy often use systems that are also referred to as “optics”, though the physics is very different. It turns out many of the same concepts of tracing charged particle's motion through a magnetic field uses similar or identical concepts as the transport of light through media with varying indices of refraction.

Recall the Lorentz Force

$\vec{F} = \frac{d\vec{p}}{dt} = q(\vec{E} + \vec{v} \times \vec{B} )$

and a charged particle left in a uniform field will travel in a helix with radius $R$

$q B R = p$

For a “thin dipole”, or short uniform field region, the deflection of a particle can be written as

$\Delta \theta = q\frac{ B \cdot \Delta l}{p}$

A particle beam with a collection of momentum particles will separate into different trajectories which can be measured from tracking detectors (your eyes can't do this!). “Chromatic” aberrations become critical for doing particle physics.

Remarkably magnetic lenses can be constructed from relatively simple coil configurations. Consider the field made by a magnet called a quadrapole:

If you follow the right hand rule, fields at the top and bottom will deflect positively charged particles going into the page up and down, defocussing like a concave lens.

The fields to the left and right will focus the same particles inwards like a convex lens. For this particular field configuration and for a specific momentum, the fields vary radially such that they will indeed have equivalent focal lengths by the equation

$f = \frac{p}{q B a}$

where $p_0$ is called your “central momentum”, a is the quadropole aperture size, and B is the strength of the field at the edge (it's zero at the center). This allows us to build very similar magnetic systems based on the same concepts and mathematics as visible light optics.

Quadrapoles also have ``chromatic'' abberations, however usually we are interested in only a small range in momenta, so it doesn't matter. We also have the advantage that things don't have to focus perfectly because tracking detectors can find the paths.

Chains of quadrupoles and dipoles can create very high resolution, 1 part in 1000, momentum resolution from position alone and 1 part in 10000 correcting for aberrations using tracking detectors. One example are the Hall A High Resolution Spectrometers at Jefferson Lab in Newport News, Virginia

They rely on superconducting, high precision magnets with a tracking system which provides 100 μm position resolution.

Such high precision was to resolve with high precision and accuracy individual nuclear excited states when scattering with high energy electron beams accelerated to energies many times the proton mass.