A method is described of designing cell phone lenses that automatically results in much smoother surfaces without the usual very "wiggly" aspheric shapes.

1. A new theory of cell phone lens designs
Dave Shafer
Summary
The highly deformed and very “wiggly” aspheric lens
shapes in conventional cell phone camera lens designs are
probably due to having extremely high-order aberrations
partly compensate for some uncorrectable lower-order
aberrations, and that is because the designs are stuck in non-optimum low-order
aberration solution regions. I show an approach that is different and which
automatically results in very smooth aspherics with minimal departures and
allows a high performance design with only 4 lenses (three acrylic, one styrene).
Cell phone camera lenses have many highly deformed aspheric surfaces and
typically every surface is aspheric. In a 5 element design that means 10 aspherics.
These designs combine both a fast speed of f/2.0 or faster and a wide angle of
about 75 degrees or more. Since they are nearly diffraction-limited over the field
that means very good aberration correction, with nearly complete control over
all the 3rd
and 5th
order aberrations, as well as many of those higher-order than
that. Since there are five 3rd
order Seidel aberrations a design with 10 aspherics
has very many more variables than are needed for 3rd
-order correction. The
result is that there are very many discrete 3rd
-order solutions with aspherics that
will look exactly the same, with respect to the first-order parameters.
The first order lens configuration can be anything at all and with only 5
aspherics, instead of 10, there will always be many 3rd
-order solutions that look
the same to us in a lens drawing. I think it is deceptive to see some 4 or 5
element designs that have the same first-order configuration and then assume
that they are basically the “same” design because they look the same to us. They
could have very different aberration contributions going through the designs,

2. because of the amounts and signs of the aspherics, even though they may have
the exact same radii, airspaces, etc. So the question for a designer is how to find
the “best” arrangement of aspherics in a design when there are so many more
than are needed for 3rd
-order correction.
The answer – of course - is that we need to look at the higher-order
aberrations, specifically the 5th
-order, and try to find which of the very many 3rd
-
order solutions is the best one to build upon when we use the higher-order
aspheric terms to control the 5th
and higher aberration orders. And here is where
it gets interesting. There is no guarantee that a particular 3rd
-order solution, using
the 4th
power aspheric terms, can be given complete control over all 9 of the
independent 5th
-order aberrations simply by using all the 6th
-order aspheric
terms. The “right” type of 3rd
order solution, not just any arbitrary one, is needed
in order to correct for all the 3rd
and 5th
order aberrations, using the 4th
and 6th
power aspheric terms.
The proof of this is easy to find. I took a 5 lens configuration that I know has
very good correction – shown below - for fast speeds and wide angles, with no
aspherics, and added an aspheric to all the surfaces. It is possible to correct all
the 3rd
and 5th
aberrations to zero using 10 aspherics but it is not a trivial exercise
to get there. The
solution space is very
non-linear. The new
design looks basically
the same as the design
without aspherics. Yet
a 6 element Double-
Gauss design does not
seem to be able to be
corrected to zero for
all the 3rd
and 5th
-
order aberrations

3. despite aspherics on every surface. So it seems that both the right first-order
configuration and the right 3rd
-order solution are needed to allow aspherics to
correct all the 5th
-order aberrations. That is, of course, not sufficient for these
fast speed wide-angle designs but it is necessary.
Once I had a good 3rd
and 5th
-order solution with 10 aspherics on 5 elements I
tried seeing if a 4 lens solution is possible. It is and this is how it looks, shown
with f/2.0 and only a 30 degree field. This is an exact solution with all the 3rd
and
5th
-
order aberrations = 0 plus a very few rays are also corrected to control some of
the 7th
-order aberrations. But there are still more variables available than are
needed. So there are multiple solutions (local minima) possible, although they
Acrylic, styrene, acrylic, acrylic

4. may look very similar. I think that there is a three element aspheric design
possible (with all 3rd
and 5th
order aberrations = 0) consisting of two positive
lenses and a negative field flattening lens, but it can’t be corrected for color with
just 3 lenses, while this 4 lens design can be with no extra lenses.
This 4 lens design only uses 4 th and 6th
power terms on the aspherics. You
can see that the aspheric lens shapes are very smooth and only the very last
surface is quite deformed. The next step was to drop the 3rd
and 5th
-order
correction, only optimize rays, and use the 8th
and 10th
order aspheric terms on all
the surfaces. By gradually increasing the field angle while using these new
variables the result was this f/2.0 and 60 degree field design that is diffraction-
limited monochromatically over the field, shown below.
I have also added in the requirement that the chief ray have an incidence angle

5. at the image of less than 30 degrees. Notice, again, the very smooth well
behaved aspheric surfaces. I am not controlling that in any way. Where I am at
right now is this next design below, where I have added the 12th
and 14th
order
aspheric terms and now have a 70 degree field that is diffraction-limited at f/2.0,
with just 4 lenses! The length is quite short – 5.0 mm for a 4.0 mm focal length.
My conclusion from this study is that the usual cell phone designs with very
wiggly aspherics are using extremely high-order aberrations to compensate for
uncorrectable lower-order aberrations – due to a non-optimum 3rd
and 5th
order
distribution inside the designs. By contrast I have started out with a very good 3rd
and 5th
-order solution before doing any ray optimization. The aspheric surfaces
stay well behaved with no wiggles because I don’t need extremely high order
aberrations to compensate for some uncontrollable lower order aberrations.
f/2.0 and 70 degree field