Single surface optics and definitions

We will define an optical system as a system which collects light; usually, the system will also make images. This requires the bending of light rays, which is accomplished using lenses (refraction) and/or mirrors (reflection).

The operation of optical systems is given by Snell's law of refraction:

$$n \sin i = n' \sin i'$$

where $n$ are the indices of refraction, $i$ are the angles of incidence, relative to the normal to the surface. For refection:

$$i' = - i$$

An optical element takes a source at s and makes an image at s'. The source can be real or virtual. A real image exists at some point in space; a virtual image is formed where light rays apparently emanate from or converge to, but at a location where no light actually appears. For example, in a Cassegrain telescope, the image formed by the primary is virtual, because the secondary intercepts the light and redirects it before light gets to the focus of the primary.

The image will not necessarily be a perfect image: all rays regardless of height at the surface, $y$, may NOT cross at the same point. This is the subject of aberrations, which we will get into in a while. Obviously, the degree of aberration will depend on how much the different rays differ in $y$, as well as depend on the shape of the surface. We define paraxial and marginal rays, as rays near the center of the aperture and those on the edge of the aperture. We define the chief ray as the ray that passes through the center of the aperture. To define nominal quantities, however, we consider the paraxial regime. In this regime, all angles are small, aberrations vanish, and a surface can be wholly specified by its radius of curvature R.

The field angle gives the angle formed between the chief ray from an object and the z-axis. Note that paraxial does not necessarily mean field angle=0. One can have an object at a field angle and still consider the paraxial approximation.

Note also that for the time being, we are ignoring diffraction. But we'll get back to that too. We are considering geometric optics, which is what you get from diffraction as wavelength tends to 0. For nonzero wavelength, geometric optics applies as scales $x\mathrel{\vcenter{\offinterlineskip \hbox{$>$} \kern 0.3ex \hbox{$\sim$}}}\lambda$.

We can derive the basic relation between object and image location as a function of a surface where the index of refraction changes (Schroeder, chapter 2).

$${n'\over s'} - {n\over s} = {(n'-n) \over R}$$

The points at $s$ and $s'$ are called conjugate. If either $s$ or $s'$ is at infinity (true for astronomical sources for s), the other distance is defined as the focal length, $f$, of the optical element. For $s=\inf$, $f= s'$.

We can define the quantity on the right side of the equation, which depends only the the surface parameters (not the image or object locations), as the power, $P$, of the surface:

$$P\equiv (n'-n)/R = n'/f' = n/f$$

We can make a similar derivation for the case of reflection; in fact, we can treat reflection by considering refraction with $n' = -n$.

$${n' \over s'} + {n' \over s} = {(n'+n') \over R}$$

$${1\over s'} + {1\over s} = {2 \over R}$$

This shows that the focal length for a mirror is given by $R/2$.

We define the focal ratio to be the focal length divided by the aperture diamter. The focal ratio is also called the F-number and is denoted by the abbreviation $f/$. Note $f/10$ means a focal ratio of ten; $f$ is not a variable in this! The focal ratio gives the beam ``width''; systems with a small focal ratio have a short focal length compared with the diameter and hence the imcoming beam to the image is wide. Systems with small focal ratios are called ``fast'' systems; systems with large focal ratios are called ``slow'' systems.

The magnification of a system gives the ratio of the image height to the object height:

$$h'/h = (s'-r) / (s-R) = n s' / n' s$$

The magnification is negative for this case, because object is flipped. The magnification also negative for reflection: $n' = -n$. Magnification not relevant for source at infinity, but still it is a very important quantity for multi-element systems.

We define the scale as the motion of image for given incident angle of parallel beam from infinity. From a consideration of the chief rays for objects on-axis and at field angle $\alpha$, we get:

$$\tan \alpha \approx \alpha = {x\over f}$$

or

$$\mathrm{scale}\equiv {\alpha\over x} = {1\over f}$$

In other words, the scale, in units of angular motion per physical motion in the focal plane, is given by $1/f$.