Design Objective:

Design a fixed-focal-length imaging lens for a camera with the following specifications:

1.Sensor format: Full-frame (36 × 24 mm)

2.Horizontal field of view: 40°

3.Maximum f-number: f/8

4.Object distance: Infinity

5.Operating spectrum: Visible light (486–656 nm)

The objective is to develop a multi-element lens system that meets these specifications while minimizing optical aberrations and maintaining good image quality across the entire sensor.

Tasks:

1. Lets start with using first order optics to calculate the focal length and entrance pupil diameter

2. Narrow down on starting number of elements and materials

3. Find out what radii and thicknesses as well as any air gaps if needed

4. Design the system, run optical ray tracing, calculate the aberrations, and image quality (RMS spot, ray fan etc)

5. Optimize the design to improve the aberrations and image quality in the field of view

First Order Calculation of focal length and EPD:

We first calculate the sensor diagonal:

\[D=\sqrt{36^{2}+24^{2}}=\sqrt{1872}=43.27\,\mathrm{mm}\]

Now we use the sensor horizontal dimensions and fov to calculate the focal length:

\[f=\frac{sensor}{2\tan(\theta_D/2)}=\frac{36}{2\tan(40^{\circ}/2)}\approx49.45\,\mathrm{mm}\]

To calculate the EPD, we use the following f/# and the above calculated focal length:

\[\mathrm{EPD}=\frac{f}{F/\#}=\frac{49.45}{8}\approx6.18\,\mathrm{mm}\]

Narrowing on the number of elements and materials:

Here we have off-axis fields in addition to polychromatic spectrum, we would thus have the following aberrations -> chromatic aberrations (longitudinal and transverse), sherical aberration, coma, astigmatism, field curvature and distortion. We need to minmiize all of them. In a very hand-wavy way, we need to satisfy 8 equations (7 aberrations + 1 focal length), thus we need atleast 8 independent variables. A triplet can give us those 8 variables. So lets start with 3 element lens design.

Now we know that for coma and distortion can be minimized with making the two extreme lenses ie first and third lens mirror identical as a seed and to have the aperture stop at the second lens. So lets go with that as out first order deisgn.

Also to minimize chromatic aberrations, let have the two extreme lenses be made of crown glass and the middle lens to be made of flint.

In addition, the  middle flint with negative power can do one more thing. Its negative power reduces the Petzval sum, which flattens the field. All the above are characterstics of classic cooke triplet.

As can be seen, we have already utilized many of those independent variables. However, this will only be a seed so all the variables are still available as independent degrees of freedom.

Initial focal lenghts:

We know the net focal lenght from above and we can use the net axial chromatic aberration to be zero. These two together give startign focal lenghts of the three lenses. We only need two equation despite three powers because we have assumed as our seed that power of first and third lens is equal.

\[\phi_1+\phi_2+\phi_3=\phi_net\]

\[\frac{\phi_1}{V_1}+\frac{\phi_2}{V_2}+\frac{\phi_3}{V_3}=0\]

We can then use these powers to calculate the seed radii of the three lens using thin lens formula.