The solution is simple, obvious, but ultimately a compromise. I have to work with the materials which I know work well, which in this case is PMMA. Rather than have a 2 material system, I am going to make the entire lens from acrylic. Switching the material of the centre element could potentially ruin the design. Not only does it change the way in which the rays trace, it also causes heavy colour aberrations due to the sudden lack of correction (the main reason for using two materials in the first place). However, I'm short on time and patients, so I am going to switch the material and re-optimised the system, making sure to lock key variables of components that are already made (front and back elements, aperture, housing, element thicknesses and spacing). This means that the new element can drop straight into the existing system and hopefully make it work.
Swapping materials
Above we can see two ray traces. The one on the left is the ray trace for the finished lens using PMMA outside elements and a polycarbonate inside element. On the right we have the same system except with the material for the centre element switched from polycarb to PMMA. You can see in the diagram how drastically it alters the ray tracing. Although the rays still converge fairly nicely, the biggest problem is the massively reduced back focal length (BFL). All the rays converge on each other at around 32mm away from the back of the final lens surface. If I were to make this lens and mount it on my camera, the mirror would smash into the rear element - disaster. Therefore, the system needs to be tweaked to bring the BFL back to an acceptable length to provide adequate clearance.
Here we can see the original system being compared to the newly optimised all PMMA system. To achieve this I locked all the variables within the system apart from the centre element surfaces, and locked the BFL back to 45mm. This provoked the optimisation process to adjust the curvatures of the centre element so that the rays converge properly on the image plane. At first I thought this was all I had to do. I was surprised that the change in material hadn't ruined the system more. However when I ran an image simulation I ended up with this:
Centre element reoptimised, everything else locked.
Unfortunately we can see the entire image has become soft. It is clear that the rays aren't converging quiet right, and as a result the image looks just out of focus. Having already played my hand and adjusted the only variables open to me without remaking components, I had to think about changing the rear element as well. As the rear element is being fed rays at a different location to what it was expecting, I need to adjust this back element as well. Reoptimisation with these settings resulted in a much sharper image across the entire frame, as shown below:
Centre and rear element reoptimised, everything else locked.
This is a good level of sharpness, which is the key improvement I want to see from making a multi element prototype. It features very little coma and spherical aberrations. Therefore, this is the system I am going with, meaning I will have to recut 2 of the 3 lenses, and directly replace them into the already made housing.
The trade off
The biggest trade off of making a lens out of only one material is resultant increase in colour aberrations. In professional lenses, many different types of glass, or even specially grown crystals are used within a single optical design to reduce the effect of colour aberrations such as colour fringing. What causes the problem though? Below we can see a tight zoom of the point where 0 degree rays converge onto the image plane. The different colours represent the 3 main wavelengths used for development ( F , d , C - Visible Light). As you can see, they don't all land on the same point. As each wavelength of light represents a colour, this shows us that that colour is splitting, a problem caused by refraction through the optical surfaces of the system.
If you struggle to see whats going on in the ray trace above, I have included a spot diagram below which shows the problem very clearly. On the left we can see the spot trace for the original system, and on the right for the new system. The colours again represent wavelength. On the left you can see the wavelengths all remain fairly tight together as they hit the image plane. This is the ideal situation, and is a result of the polycarbonate element in the centre of the system reversing some of the effect of the refraction occurring in the PMMA elements. On the right we can see the blue wavelength is spread out much wider than red and green. This will be visible in photos, and will be seen easiest along contrasty edges. All in all, polycarbonate was included in the original design to avoid this error, and without it we are unfortunately likely going to see chromatic aberrations across the frame.
