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. We start with the standard cooke triplet deisgn. This standard cooke triplet design would have N-SK16 as the crown glass and F2 as the flint glass.

Initial focal lenghts:

We know the net focal length 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. Seed thickness and air gaps are taken to be 10% of radii and focal length respectively. Once we have all the data, system is designed and exact ray tracing is done.

Results:

The following polychromatic spot diagram for each field is at its own best focus. That focus might not be at the same plane. The sensor, however, is at a fixed location and this the rms spot for each field needs to be done at the plane.

It should also be noted that at the best focus for each field (not on the same plane), the residual aberrations e.g. spherical. coma, astigmatism are all below 0.1 waves. Zernike aberration values in terms of waves are shown below. In additoin, distortion at the sensor plane (chosen to be the best focus plane of on-axis field at d line) is also small ~ 1.4% for largest off-axis field.

At the best focus for each field, defocus is very minimal and there to mostly compensate for spherical aberration. Chromatic Aberration - axial as well as transverse for on-axis field is shown. The transverse spread is not much, the lateral spread is ~ 0.31 mm. This spread remains around the same value for all the fields.

Values above are shown for each field at its best focus - which does not lie on the same plane. This means there is field curvature in the system. It is shown below. Left panel shows almost a field curvature of 1.7 mm for the largest field. Thus at the sensor plane, this FC shows up as zernike defocus (right panel).

The polychrhromatic spot diagram shown above was at each field's best focus. Below, we see the comparison between the rms spot radius as well as rms wavefront error for polychromatic (weighted equally) for the two cases - when the fields are at their best focus and when they fields are imaged at the sensor plane.

The polychromatic rms spot radius values for comparison are for field {0,10,20} changes from {6.9, 7.2, 10.9} µm for best focus case to {6.9, 24.3, 80.5} µm for sensor plane. Similarly, The polychromatic wavefront error values for comparison are for field {0,10,20} changes from {0.068, 0.072, 0.121} waves for best focus case to {0.068, 0.407, 1.467} waves for sensor plane

The gap comes from field curvature (as seen before using aberrations)

For each field's best focus the residual aberrations - spherical, coma, astigmatism are all below ~0.1 waves. Therefore, on a flat sensor, the perfomance is limited by field curvature (1.66 mm for the 20 degree field), which a field flattener can correct. Thus, our three element design can only get us so far. Optimizing this system usig our in-built custom tracer did not move the deisgn a lot as it is already optimized by using the cooke's triplet arrangement of equiconvex lens and near-symmetric stop position.

Remedy for Field Curvature:

To add the fourth element: summing up the petzval sum of the elements in this system, we can add a fourth element that makes the net petzval sum to be zero. This will give us the power/refractive index we want out fourth element to have. Wanting the power of this lens to not be very high (so that our other aberratiosn are controlled), we would choose a smaller n glass. Once we have the power, we can design an equi-convex lens near the image plane and go from there.

Conclusions:

The lens is near diffraction-limited in its monochromatic aberrations. The f/8 full-frame triplet (N-SK16 crowns / F2 flint, plano-convex bending, near-central stop) corrects spherical, coma and astigmatism to under ~0.1 waves at every field, giving per-field RMS spots of 7–11 µm against a 5.7 µm Airy radius.

On a flat sensor the design is field-curvature (Petzval) limited. The uncorrected Petzval sum curves the image surface ~1.66 mm , which turns the tiny per-field blur into a ~80 µm blur on a flat-sensor. This is an inherent limit of a three-element triplet,; removing it needs a field flattener.

Distortion and chromatic aberrations are well-controlled. Flat-sensor distortion stays ≤ ~1.4% (mild pincushion), and lateral colour is small — both benefits of the near-symmetric stop, which cancels the odd aberrations. Axial colour is held by the achromatic power split (Σφ/V = 0). The result is a compact, manufacturable triplet whose sole significant limitation is the classic triplet field curvature.