A "spherical lens" is a lens whose surface has the shape of
(part of) the surface of a sphere. On this page, we will
determine some values for the focal length of a spherical lens.
Figure 1  Spherical Aberration

A spherical lens doesn't actually bring parallel rays to a common focus
(see figure 1). Rather, the "focus" of the lens depends on the
distance from the distance from the center of the lens, with light
passing through the edges of the lens coming to a focus closer to the
lens than light passing through the center. This produces the
phenomenon of "focus shift", which is well known to
photographers: If your lens suffers from severe "focus shift",
and you focus the image with the diaphragm wide open, and then stop
down to take the picture, the picture won't be in focus. With the
lens opened wide the image is dominated by light passing through the
lens far from the center; consequently, it's the
edge rays
which you've adjusted the lens to bring to a sharp focus. The
center rays, on the other hand, are coming to a focus
behind
the film plane; when the lens is stopped down, only those (out of
focus) center rays are left.
On the remainder of this page, we'll find an exact formula for the
focal length of a spherical lens, and we'll then find an approximate
formula for the focal length at the
center of the lens.
This problem is somewhat messy, and
we're going to break the problem down into easier to handle pieces by
cutting the lens in half. We'll find the focal length of each
half
lens and then put them together to get the focal length of a
convexconvex spherical lens.
Part I: PlanoConvex HalfLens
Figure 2
 PlanoConvex Half
Lens

We'll start by looking at a lens which is spherical on one surface and
flat on the other, with parallel rays of light entering on the
flat
side. Since the light is entering perpendicular to the flat
surface, we can ignore the effect of the planar side of the lens, and
just analyze the convex surface.
Before we begin, we should point out something that may not be
obvious: This is not an ideal lens, and the focal length when
parallel rays enter the flat side may
not be identical to the
focal length when parallel rays enter the convex side! The light
is following different paths in those two cases  the paths have not
just been reversed  and there is no
a priori guarantee that
the results will be simple mirror images of each other. Obviously
the case where the light enters the flat side is simpler to analyze, as
we only need to deal with a single refracting surface, and that is
where we are starting.
In figure 3 we've shown the lens as a full halfsphere, with the center
of the sphere at the origin. The flat side of the lens cross
section lies on the
x axis, and the axis of the lens
lies on the
y
axis. For each ray passing through the lens, we can define the
"focal point" of that particular ray as being the point at which the
ray crosses the axis of the lens.
Figure 3  Path
Followed by a Single Ray

In figure 3 we've shown a single ray passing through the lens, with its
focal point at the point marked "
focus". We'll now
determine how far from the lens the focal point lies.
A real lens has nonzero thickness, unlike an ideal lens, and so the
"focal length" of a real lens is a little ambiguous. In figure 3
we shown two "focal lengths":
f is the distance
from the front of the lens to the focus, and
F is the
distance from the center of the sphere to the focus.
The point at which the ray passes through the surface of the lens is
(x,y).
It's at distance
R from the origin (since the lens
surface is spherical). The radius line, labeled
R,
is perpendicular to the surface of the lens at
(x,y).
We've also shown the line tangent to the lens surface at
(x,y);
that line is, of course, perpendicular to the radius line.
We can find
y as a function of
x.
We want to find
f as a function of
x.
To do that, we'll first find the tangent of angle ξ, and then use that
to find the length marked
F  y in figure 3; from that
we can determine both
f and
F.
The ray hits the surface of the lens at angle
θ_{0} with
the line perpendicular to the surface, which is also the angle between
the radius line and the
y axis, and is also the angle
between the line tangent to the lens and a horizontal line. We
can read off directly:
1)
The ray emerges from the lens at angle
θ_{1} with a
line perpendicular to the surface of the lens. In figure 3, we've
shown the refractive indices of the
lens and the
surrounding
environment as
N_{l}
and
N_{e}. From
Snell's Law, we have
2)
We can see from the figure that ξ =
θ_{1} 
θ_{0}. From that and
the formula for the sine of the sum of two angles (which is derived,
somewhat haphazardly,
here), we
have
3)
which we can clean up a bit to produce
4)

Note that this is
not defined for all possible values of
x.
If x>R then the first square root is imaginary. That's not a
problem; x>R implies we're looking at a point outside the lens!
However, if x/R > N
_{e}/N
_{l}, then the second
square root is imaginary. What does that mean? In that case
sin(ξ) is also imaginary. We can't have light emerging from the
lens at an
imaginary angle, of course. What's going on is
that the ray is hitting the inside of the lens at a steeper than
critical angle, and it's being totally reflected inside the lens.
Once we exceed the angle of TIR nothing comes out  the surface of the
lens acts as a perfect mirror. At the critical angle, sin(
θ_{1})=1,
and the ray comes out traveling tangent to the lens surface; beyond the
critical angle the ray would actually be bent back inside the
lens. That doesn't happen of course; instead it's just reflected.
We'll refer to the value of
x/R at which the ray hits
the lens surface at the critical angle as the
cutoff value for
x.
The effective diameter of the lens is actually the cutoff value; the
parts of the lens outside that distance aren't doing anything for us.
We've found sin(ξ). What we want, however, is the tangent of
ξ. We won't write that formula out in full; we'll just say:
5)
which is all we'll actually need later on.
Finally, looking back at figure 3, we can find the value of
f
from the triangle with (x,y) at its upper vertex and the focus at one
of its lower vertices:
6)
By dividing through by
R we obtain a slightly more
useful form, as it doesn't depend on units we use to measure the
diameter of the lens:
7)
Using a Python program (
here)
we've found
f/R between x=0 and the point at which
cutoff occurs. In figure 4 we show a plot of
f/R
versus
x/R for a range of values of
N=
N_{l}/
N_{e}
running from 1.1 to 5. As
N approaches 1 the focal
length (at the lens center) goes to infinity. As
N
approaches infinity, the focal length approaches 0. At all focal
lengths, as we move farther from the lens center the focal length
decreases.
Figure 4 makes it clear that the focal length varies with the distance
from the lens center. It also shows graphically the effect of
lens cutoff, which limits the maximum size of lens of this sort which
we can actually build. For glass with a refractive index of
roughly 1.5, we see that the focal length at the center of the lens is
about twice the radius of curvature, and cutoff is at about 0.67 times
the radius of curvature. The
average focal length of the
lens will be quite a bit shorter than the focal length at the center,
and probably falls between 1 and 1.5 times the radius of
curvature. The lens "speed", as an F number, is the focal length
divided by the diameter. If the average focal length of our
"maximum" lens is as small as 1, with cutoff at about 0.67 the lens
size will be about F/0.75.
Finally, we'll find the focal length at the
center of the lens,
by taking the limit of everything as
x goes to
zero. From equation (
4) we can see that
8)
We define
9)
Note that, for the situations we're considering, n≥1.
We can write now (8) as
10)
As ξ goes to zero, tan(ξ) approaches sin(ξ), and we can finally write
11)

That's the focal length at the center of the lens, and, as we
shall see later, it is also
twice the (approximate) focal length of any
small
convexconvex lens, such
as a magnifying glass.
As we've also seen, the cutoff angle is the point where sin(
θ_{1})
= 1. From equation (
2) we see that this is:
12) 
And finally, we'll find the focal length at the cutoff point.
From (4) and (12) we have:
13)
and then from (5) we have:
14)
and finally, from (7) and (14), we obtain
15)
Part II: Infinitely Thick Convex Half
Lens
Figure 21:
Convex Halflens

We're now going to reverse the lens, and find the focal length when
parallel rays enter the convex surface. However, we're not going
to allow the rays to come out through the flat side  we're going to
assume the lens is so thick that the rays come to a focus
inside
the glass. In other words, we're finding the focal length
when the light enters an
arbitrarily thick convex halflens.
We'll find this result useful by itself when we eventually look at how
eyes work, and we'll also find it interesting to contrast the result we
obtain here with the result we obtained in Part I.
In figure 22 we've shown the lens as a full halfsphere, with the
center
of the sphere at the origin. The axis of the lens lies on the
y
axis. For each ray passing through the lens, we can define the
"focal point" of that particular ray as being the point at which the
ray crosses the axis of the lens.
Figure 22:
Path followed by a single ray

In
figure 22 we've shown a single ray
passing through the lens, with its focal point at the point marked "
focus".
We'll now determine how far from the surface of the lens the focal
point lies. We'll proceed very much as we did in
Part I above.
We will continue to define
n as in equation (
9) above:
21)
From figure
22 we can see that
22)
From
Snell's Law we have
23)
From figure
22, where ξ is the angle
between the ray and the
y
axis after it enters the lens, we have
24)
And so we have
25)
which, using (
21), we can write as
26) 
With equation (
5) that gives us tan(ξ). From
figure
22 we can see that
27)
or, in terms of
x/R,
28) 
As we did with light from the other direction, we've used a Python
program (
here) to find the
focal length for all distances from the center of the lens for a number
of refractive indices. We once again see that, as we move away
from the center of the lens, the focal length decreases.
Note that, for all refractive indices, the plot extends all the way to
the edge of the lens (or it should; the program seems to have left off
the last point on each curve). There is no
cutoff when the light enters the
curved side of the lens. When moving from a less dense medium to
a more dense medium there is no "critical angle"  light can enter the
denser medium at any angle.
Finally, we'll find a formula for the focal length near the center of
the lens. First we find the limit of sin(ξ) as we approach the
center of the lens by dropping out the terms in x
^{2}:
29)
We know that, as
x
approaches 0, tan(ξ) approaches sin(ξ). So, we also have
210) 
The NonIdeal Nature of a Single
AirGlass Interface
Very near the center, the curvature of the lens can be fully
characterized by the second derivative of the function which
describes its surface. In other words, very near the center, it
doesn't matter what the actual figure of the lens is; viewed through a
sufficiently small "pupil" all lenses act like spherical lenses.
Consequently, results obtained for the limiting case of a small lens
will apply to
all physical
lenses, not just spherical ones.
We have just seen a clear demonstration that a single airglass
interface can
not form an
ideal lens. With an ideal lens, rays traveling parallel to the
lens axis are brought to a single focus at a fixed distance from the
lens. But when we consider an airglass interface which has been
formed into a lens, we realize that's not true: Comparing
equations (
11) and (
210)
we see that parallel rays passing
into
the glass are brought to a focus
n
times farther from the lens than parallel rays passing
out of the glass. (This
doesn't violate the "reversibility of light paths" because the paths
followed in the cases are very different  parallel rays passing in
are traveling parallel in the air and are converging in the
glass. Parallel rays passing out are traveling parallel in the
glass, and converging in the air. The "reverse" of the first case
is a point source located in the glass being focused to parallel rays
as it passes into the air, and the "reverse" of the second case would
be a point source in the air being focused to parallel rays as it
passes into the glass, and in these cases the light would indeed
precisely "backtrack" the rays in the corresponding "converging" case.)
Furthermore, as we have shown
here,
rays passing through an ideal lens are deflected according
to the formula
211)
where
x is the distance from the
lens center. In particular, this means rays passing through the
center of the lens are not
deflected. That's patently false for a single airglass surface,
so it should not surprise us that such a surface can't form an ideal
lens.
As we shall see later, the story is quite different when there are
two glassair surfaces, as in an
ordinary camera lens, or a magnifying glass.
Part III: Combining Two HalfLenses
We've found two different values for the focal length of half of a
lens. We'd now like to combine them by gluing two half lenses
together.
Figure
31: Two
lenses back to back

For us to say anything intelligent about the result of our "glue job",
the light passing through the lens must follow paths we've already
analyzed. In particular, for light coming
out of the lens (passing out
through the "second" surface), the light must either be traveling
parallel to the lens axis (
figure 2), which is
what we analyzed in
Part I, or it must be
following lines leading directly away from the "inside" focus of the
lens
(figure 21, with arrows reversed),
which is the path we analyzed in
Part II.
We can exhibit one case in which such a glued up lens can be understood
with the results of Part I: If we place a point source at the
focus of a halflens, light
emerging from the lens will be traveling parallel to the lens
axis; if that light then enters a second halflens, it will
converge on the focus of that halflens. This is identical to the
case in which two ideal lenses are placed next to each other and light
originating at the focus of one travels to the focus of the
other. We have shown this arrangement in figure 31.
In this case we have an object (point) at distance
f in front
of the lens, and it's forming an image at the same distance
behind the lens. To the extent that this combination is acting as
an ideal lens, this implies that the focal length of the lens must be
f/2
(see our ideal lens images page,
case 1). Note,
however, that we have ignored the thickness of the lens body, which is
going to throw this result off a bit  and we've also ignored the fact
that a spherical lens hasn't got a particularly well defined focal
length to start with. This result will hold for small, thin
spherical lenses, but really thick spherical
lenses will be far from ideal.
Applying this to equation (
11), and recalling our
definition of
n from equation (
9),
we conclude that the focal length of a small, symmetric convexconvex
spherical lens must be
31)

As we will see later, this matches the focal length of a
tiny convex lens,
which gives us confidence that the model here is accurate.
A Small, Symmetric Convex Lens
Of course, when we glue together two
large spherical lenses,
the result is going to have severe "spherical aberration": The
focal length will be shorter near the edge than it is in the center,
with the result that parallel rays falling on the entire lens won't be
brought to a precise focus. However, if we glue together two
small
lenses, then the result will bring light traveling parallel to the axis
to a (nearly) precise focus. But will such a lens actually behave
as an ideal lens does? Will it focus a distant scene onto a flat
focal plane? To answer this question we need to know more about
how the lens will focus light which is
not arriving parallel to
the axis.
For scenes which are
nearly onaxis, the answer is yes, it will
behave as an ideal lens. For scenes which extend a substantial
distance offaxis the answer is more complex. The analysis is
somewhat lengthy, however, and I've moved it off to a separate page on
tiny convex lenses.
Page
created on 01/15/2009. Last updated 1/18/2009.