The concepts already introduced are sufficient for our purposes.
What remains is to apply Snell's law for the geometry in question and
derive some formulas that will be needed later.
In this section the so-called lensmaker's formula (equation 
)
and the Gaussian lens formula (equation )
will be derived. They are useful when dealing with thin lenses and are used
in calculations in section . Some of the short-cuts and
approximations in the derivation give rise to differences between theory and
practice as will be discussed in section .
Therefore, the derivation is included in some detail here.
Consider two media with different indices of refraction. What shape should the interface between the two media have to give a perfect image according to approximate theory? The source do not have to be static so the time the rays use to travel from the source (S) to the observation point (P) has to be independent of where the ray hits the surface. The interface satisfying this restriction is called a Cartesian oval and was discovered by Descartes (see cartesianoval).
The optical path length (
) is the distance the light would travel
in vacuum during the time it travels in the system.
This length should be independent of which point on the surface (A)
the ray hits.
OPL & = & n_1 l_1 + n_2 l_2
& = & n_1 s_1 + n_2 s_2
n_1 l_1 + n_2 l_2 & = & constant.
Here 
and 
are the indices of refraction of the media.
and 
are the shortest possible distance the light
can travel from S to P, and 
, and 
are two other
distances for an arbitrarily chosen point (A) on the interface
between the two media.
It can be shown that conic sections fulfill these
criterion, but it is too difficult to manufacture such shaped
interfaces for them to be of any practical interest.
The easiest curved surface to manufacture is the spherical
interface (see spherical). Most lenses have spherical
shaped surfaces and because of this the spherical geometry is
discussed here.
Consider the spherical interface in spherical and apply Snell's law.
n_1 
& = & n_2 
n_1 (+ ) & = & n_2 ( -)
This can be rewritten using trigonometric formulas,
n_1 (+ ) & =
& n_2 (- ) \;.
If we then substitute for the angles and rearrange we get,
n_1 (hl_1 R-xR + s_1+xl_1 hR) & =
& n_2 ( hR s_2-xl_2 -R-xR hl_2)
n_1 R+s_1l_1& = & n_2 s_2-Rl_2
n_1l_1+n_2l_2 & = & 1R(n_2 s_2l_2
-n_1 s_1l_1).
If A is moved the ray no longer converges to P, but
to somewhere in the vicinity. If the surface is shaped as
a Cartesian oval the rays would still converge towards P.
If A is restricted to be very close to the vertex 
will be small. A series expansion of sin to the first order in
gives
l_1 = s_1 & and & l_2 = s_2 \;.
Consequently from equation
n_1s_1 + n_2s_2 & = & 1R( n_2 - n_1) \;.
As long as the assumptions are fulfilled, P will act as a focal point for
rays from S. If S is imaged at infinity, the distance is said
to be the first focal length or object focal length of the system.
Similarly if is imaged at infinity, the second focal
length, or the image focal length are found (see convergenlenses).
That is ,
s_2 &
& f_1=s_1= n_1n_2-n_1 R\;,
s_1 & & f_2=s_2= n_2n_2-n_1 R \;.
At this point negative curvature is introduced. This is spherical
surfaces as shown in divergenlenses with the center of curvature in
the medium with smallest index of refraction.
When the light passes through such a surface it will spread out.
If an object located far out is seen through such a surface, the object
will appear closer, as shown in the left part of divergenlenses.
A point like this, towards which the rays converge, is called a virtual image point.
Also virtual object points 
, as shown in the right part of
divergenlenses exist. These are points where an object located infinitely long
away in the second medium appears to be when viewed from the first medium.
If 
, the radius of curvature, is interpreted
negative for these systems with the center of curvature in front of the interface
relative to the ray direction, the same formulas apply
and the focal length also becomes negative.
This is important for more complicated systems such as one-lens systems where two spherical surfaces are combined to reshape the wavefront. The shapes of the interfaces depend on the function of the lens and the material it is made of (its index of refraction etc.). The formula which describes this geometry, under the assumptions made, is the lensmaker's formula, which will be derive here.
In lens the rays are emitted from point S and converge
towards P. Let us first consider the front surface of the lens.
If the circle at the right in lens thought of as being compact,
the optical system is similar to the one in spherical.
For this system formula applies, this gives
n_1s_11 + n_2s_21 & = & n_2 - n_1R_1.
If on the other hand the left circle had been compact the
rays would have looked as if they came from another point, P', to the left of S.
Applying formula on this surface gives
n_2s_12 + n_1s_22 & = & n_2 - n_1R_2.
From lens is seen 
.
Inserted in equation this gives
n_2d-s_21+n_1s_22 & = & n_2 -n_1R_2 \;,
n_2s_11+n_2s_22& = & (n_2-n_1)(1R_1-
1R_2) - n_1 ds_21(d-s_21) \;.
If the lens is thin (
small) this reduces to the lensmaker's formula,
1s_1+1s_2 & = & (n_2n_1-1)(1R_1-1R_2).
For a given lens in the limit 
the right hand side of
equation is a constant with dimension 
.
If is made large equation is recovered.
The left hand side of equation is then just the inverse of
the object focal length. If on the other hand is made large,
the left side of the equation is the inverse of the image focal length as in
equation . For a lens this is simply called
the focal length, 
, which gives the Gaussian lens formula.
1s_1+1s_2 & = & 1f