A Trip to Riga and Its Consequences
When my friend Burkhard, also a Minox Riga enthusiast, traveled to Riga this summer, tracking down the VEF Minox was at the top of his list. At my request, he also searched for the design drawings for this camera’s lens, the Minostigmat. I intended to use it for a Minox lens calculation.
And he found what he was looking for at the VEF History Museum. With the help of Dace Kaprāne, the chief curator, he located them in the archives. Tucked away in a folder, he actually found what I had been looking for for a long time but couldn’t find anywhere: the original drawings of the Minostigmat.
So here, courtesy of the VEF History Museum, are the blueprints for the manufacture of the famous lens, published for the first time:
Expectations, theory, and reality
I had been searching for the exact specifications of the Minostigmat for quite some time because I was determined to calculate its imaging performance. After measuring its performance in practice in 2024, I wanted to compare the two sets of results. I was also interested in the optical parameters so that I could evaluate this lens in general.
The results I’m about to present here are impressive in two respects. First of all, the specifications show just how outstanding this lens is, even in theory. On top of that, there’s the astonishingly good correlation with my earlier practical measurements using test charts.
You can find a brief, easy-to-understand introduction to how a design engineer goes about designing a Cooke triplet here.
So that you can follow along with the calculations below on your own, I’ve broken everything down into the smallest possible steps using specific numbers. This way, you can verify all the results yourself using a calculator. While this makes the article a bit long, it ensures that everything is transparent.
Join me on this exciting journey through the process of calculating a lens. Don’t worry – the four basic arithmetic operations are all you need.
If you’re not interested in the calculations, you can go directly to the results.
Focal length
My first step was to verify the plausibility of the Minostigmat drawings and the values derived from them. A good way to do this is to calculate the focal length based on this information. The result is known: it should be 15 mm.
However, there was another problem. As you can see in the assembly drawing above, the two dimensions for the distances between the three lenses are missing. I therefore had to measure these dimensions directly from the drawing. Of course, this is subject to reading errors. Both dimensions (d2 and d4, see picture below) strongly influence the calculation of the focal length. Using my measured values, the focal length comes out to 14.73 mm. This is close to the expected value.
However, it’s safe to assume that d2 and d4 were chosen so that the focal length is exactly 15 mm. I therefore varied these two values over several iterations so that the deviation from my measured values is minimal and a focal length of 15 mm is obtained.
As a result, the value I measured for d2 changes from 1.25 mm to 1.1265 mm, and the value for d4 changes from 1.10 mm to 0.9520 mm. I then performed all calculations using these values.
The Matrix Method
It is worth noting that the elegant 2×2 matrix formalism, which is now standard in optical design, did not yet exist when Hans Schulz calculated the Minostigmat in 1935. Each refracting surface and each air gap is represented by a small matrix, and the entire system reduces to a single matrix product. The explicit matrix method for lens design was first formalized fifteen years later, in Clyde W. Harris’s 1950 paper “Optical Design by a Matrix Method”, which showed how interval and surface matrices could be multiplied together to yield the paraxial, aberrational, and even chromatic behavior of a complete system in a single, compact operation. I will use this method in the following.
The underlying mathematics, however, was far from new. The paraxial relations the matrices encode trace back to Carl Friedrich Gauss’s 1841 treatment of the “cardinal points” of an optical system, and the third-order aberration theory needed to judge image quality – spherical aberration, coma, astigmatism, field curvature, distortion – had been laid out by Ludwig von Seidel in 1857. By the 1930s, both were standard tools in German optical practice, from Zeiss to VEF.
In the calculations, I use the symbols according to standard convention:
I took the specific dimensions and the refractive indices n from the VEF drawings from 1939 shown above. The following two tables show the numerical values.
0. Input Data
| Surface | R (mm) | n (before) | n’ (after) |
|---|---|---|---|
| 1 | 4.2950 | 1.0000 (air) | 1.6116 |
| 2 | 21.0000 | 1.6116 | 1.0000 (air) |
| 3 | -12.8640 | 1.0000 (air) | 1.6228 |
| 4 | 4.2000 | 1.6228 | 1.0000 (air) |
| 5 | 10.4100 | 1.0000 (air) | 1.6116 |
| 6 | -8.6086 | 1.6116 | 1.0000 (air) |
| Gap | d (mm) | n |
|---|---|---|
| d1 (inside lens 1) | 0.800000 | 1.6116 |
| d2 (air, 1→2) | 1.126477 (measured: 1.25) | 1.0000 |
| d3 (inside lens 2) | 0.250000 | 1.6228 |
| d4 (air, 2→3) | 0.952009 (measured: 1.10) | 1.0000 |
| d5 (inside lens 3) | 1.000000 | 1.6116 |
Note: the ray-transfer vector used here carries the real ray angle u (not the reduced angle n·u), so the translation matrix is \(T=\begin{pmatrix}1 & d\\0 & 1\end{pmatrix}\) — without dividing by n.
1. Refraction Matrices R1–R6
Formula: \(C = -\dfrac{n’-n}{R\cdot n’}\), \(D=\dfrac{n}{n’}\)
$$R_1 = \begin{pmatrix} 1 & 0 \\ -0.088358 & 0.620501 \end{pmatrix}$$
$$R_2 = \begin{pmatrix} 1 & 0 \\ 0.029124 & 1.611600 \end{pmatrix}$$
$$R_3 = \begin{pmatrix} 1 & 0 \\ 0.029834 & 0.616219 \end{pmatrix}$$
$$R_4 = \begin{pmatrix} 1 & 0 \\ 0.148286 & 1.622800 \end{pmatrix}$$
$$R_5 = \begin{pmatrix} 1 & 0 \\ -0.036455 & 0.620501 \end{pmatrix}$$
$$R_6 = \begin{pmatrix} 1 & 0 \\ -0.071045 & 1.611600 \end{pmatrix}$$
2. Translation Matrices T1–T5
$$T=\begin{pmatrix}1 & d\\0 & 1\end{pmatrix} \quad \text{(no division by n)}$$
$$T_1 = \begin{pmatrix} 1 & 0.800000 \\ 0 & 1 \end{pmatrix}$$
$$T_2 = \begin{pmatrix} 1 & 1.126477 \\ 0 & 1 \end{pmatrix} \quad $$
$$T_3 = \begin{pmatrix} 1 & 0.250000 \\ 0 & 1 \end{pmatrix}$$
$$T_4 = \begin{pmatrix} 1 & 0.952009 \\ 0 & 1 \end{pmatrix} \quad $$
$$T_5 = \begin{pmatrix} 1 & 1.000000 \\ 0 & 1 \end{pmatrix}$$
3. Sequential Matrix Multiplication
$$M = R_6 \cdot T_5 \cdot R_5 \cdot T_4 \cdot R_4 \cdot T_3 \cdot R_3 \cdot T_2 \cdot R_2 \cdot T_1 \cdot R_1$$
Product rule for \(X=\begin{pmatrix}a&b\\c&d\end{pmatrix}\), \(P=\begin{pmatrix}e&f\\g&h\end{pmatrix}\):
$$X \cdot P = \begin{pmatrix} ae+bg & af+bh \\ ce+dg & cf+dh \end{pmatrix}$$
Each step below shows the multiplication in three lines: the two matrices being multiplied, then the elementwise arithmetic, then the numeric result — each continuation line starts with “=”.
Start: P0 = R1
$$P_0 = \begin{pmatrix} 1 & 0 \\ -0.088358 & 0.620501 \end{pmatrix}$$
Step 1: P1 = T1 · P0
$$P_1 = \begin{pmatrix} 1 & 0.8 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ -0.088358 & 0.620501 \end{pmatrix}$$
$$= \begin{pmatrix} 1\cdot1+0.8\cdot(-0.088358) & 1\cdot0+0.8\cdot0.620501 \\ 0\cdot1+1\cdot(-0.088358) & 0\cdot0+1\cdot0.620501 \end{pmatrix}$$
$$= \begin{pmatrix} 0.929314 & 0.496401 \\ -0.088358 & 0.620501 \end{pmatrix}$$
Step 2: P2 = R2 · P1
$$P_2 = \begin{pmatrix} 1 & 0 \\ 0.029124 & 1.611600 \end{pmatrix}\begin{pmatrix} 0.929314 & 0.496401 \\ -0.088358 & 0.620501 \end{pmatrix}$$
$$= \begin{pmatrix} 1\cdot 0.929314+0\cdot(-0.088358) & 1\cdot 0.496401+0\cdot 0.620501 \\ 0.029124\cdot 0.929314+1.611600\cdot(-0.088358) & 0.029124\cdot 0.496401+1.611600\cdot 0.620501 \end{pmatrix}$$
$$= \begin{pmatrix} 0.929314 & 0.496401 \\ -0.115333 & 1.014459 \end{pmatrix}$$
Step 3: P3 = T2 · P2
$$P_3 = \begin{pmatrix} 1 & 1.126477 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0.929314 & 0.496401 \\ -0.115333 & 1.014459 \end{pmatrix}$$
$$= \begin{pmatrix} 1\cdot 0.929314+1.126477\cdot(-0.115333) & 1\cdot 0.496401+1.126477\cdot 1.014459 \\ 0\cdot 0.929314+1\cdot(-0.115333) & 0\cdot 0.496401+1\cdot 1.014459 \end{pmatrix}$$
$$= \begin{pmatrix} 0.799393 & 1.639162 \\ -0.115333 & 1.014459 \end{pmatrix}$$
Step 4: P4 = R3 · P3
$$P_4 = \begin{pmatrix} 1 & 0 \\ 0.029834 & 0.616219 \end{pmatrix}\begin{pmatrix} 0.799393 & 1.639162 \\ -0.115333 & 1.014459 \end{pmatrix}$$
$$= \begin{pmatrix} 1\cdot 0.799393+0\cdot(-0.115333) & 1\cdot 1.639162+0\cdot 1.014459 \\ 0.029834\cdot 0.799393+0.616219\cdot(-0.115333) & 0.029834\cdot 1.639162+0.616219\cdot 1.014459 \end{pmatrix}$$
$$= \begin{pmatrix} 0.799393 & 1.639162 \\ -0.047236 & 0.674033 \end{pmatrix}$$
Step 5: P5 = T3 · P4
$$P_5 = \begin{pmatrix} 1 & 0.25 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0.799393 & 1.639162 \\ -0.047236 & 0.674033 \end{pmatrix}$$
$$= \begin{pmatrix} 1\cdot 0.799393+0.25\cdot(-0.047236) & 1\cdot 1.639162+0.25\cdot 0.674033 \\ 0\cdot 0.799393+1\cdot(-0.047236) & 0\cdot 1.639162+1\cdot 0.674033 \end{pmatrix}$$
$$= \begin{pmatrix} 0.787584 & 1.807670 \\ -0.047236 & 0.674033 \end{pmatrix}$$
Step 6: P6 = R4 · P5
$$P_6 = \begin{pmatrix} 1 & 0 \\ 0.148286 & 1.622800 \end{pmatrix}\begin{pmatrix} 0.787584 & 1.807670 \\ -0.047236 & 0.674033 \end{pmatrix}$$
$$= \begin{pmatrix} 1\cdot 0.787584+0\cdot(-0.047236) & 1\cdot 1.807670+0\cdot 0.674033 \\ 0.148286\cdot 0.787584+1.622800\cdot(-0.047236) & 0.148286\cdot 1.807670+1.622800\cdot 0.674033 \end{pmatrix}$$
$$= \begin{pmatrix} 0.787584 & 1.807670 \\ 0.040137 & 1.361738 \end{pmatrix}$$
Step 7: P7 = T4 · P6
$$P_7 = \begin{pmatrix} 1 & 0.952009 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0.787584 & 1.807670 \\ 0.040137 & 1.361738 \end{pmatrix}$$
$$= \begin{pmatrix} 1\cdot 0.787584+0.952009\cdot 0.040137 & 1\cdot 1.807670+0.952009\cdot 1.361738 \\ 0\cdot 0.787584+1\cdot 0.040137 & 0\cdot 1.807670+1\cdot 1.361738 \end{pmatrix}$$
$$= \begin{pmatrix} 0.825795 & 3.104053 \\ 0.040137 & 1.361738 \end{pmatrix}$$
Step 8: P8 = R5 · P7
$$P_8 = \begin{pmatrix} 1 & 0 \\ -0.036455 & 0.620501 \end{pmatrix}\begin{pmatrix} 0.825795 & 3.104053 \\ 0.040137 & 1.361738 \end{pmatrix}$$
$$= \begin{pmatrix} 1\cdot 0.825795+0\cdot 0.040137 & 1\cdot 3.104053+0\cdot 1.361738 \\ -0.036455\cdot 0.825795+0.620501\cdot 0.040137 & -0.036455\cdot 3.104053+0.620501\cdot 1.361738 \end{pmatrix}$$
$$= \begin{pmatrix} 0.825795 & 3.104053 \\ -0.005201 & 0.731877 \end{pmatrix}$$
Step 9: P9 = T5 · P8
$$P_9 = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 0.825795 & 3.104053 \\ -0.005201 & 0.731877 \end{pmatrix}$$
$$= \begin{pmatrix} 1\cdot 0.825795+1\cdot(-0.005201) & 1\cdot 3.104053+1\cdot 0.731877 \\ 0\cdot 0.825795+1\cdot(-0.005201) & 0\cdot 3.104053+1\cdot 0.731877 \end{pmatrix}$$
$$= \begin{pmatrix} 0.820594 & 3.835930 \\ -0.005201 & 0.731877 \end{pmatrix}$$
Step 10: P10 = R6 · P9 (= final system matrix M)
$$M = \begin{pmatrix} 1 & 0 \\ -0.071045 & 1.611600 \end{pmatrix}\begin{pmatrix} 0.820594 & 3.835930 \\ -0.005201 & 0.731877 \end{pmatrix}$$
$$= \begin{pmatrix} 1\cdot 0.820594+0\cdot(-0.005201) & 1\cdot 3.835930+0\cdot 0.731877 \\ -0.071045\cdot 0.820594+1.611600\cdot(-0.005201) & -0.071045\cdot 3.835930+1.611600\cdot 0.731877 \end{pmatrix}$$
$$= \begin{pmatrix} 0.820594 & 3.835930 \\ -0.066681 & 0.906968 \end{pmatrix}$$
$$M = \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} 0.820594 & 3.835930 \\ -0.066681 & 0.906968 \end{pmatrix}$$
(High-precision computation gives \(M=\begin{pmatrix}0.820630 & 3.836059\\-0.066663 & 0.906960\end{pmatrix}\) — small differences above are rounding artifacts from carrying only 6 decimal places through the chain; both check against det(M) ≈ 1.)
Check: det(M) = A·D – B·C ≈ 1.000 ✓
4. Results
Effective focal length
$$f_{eff} = -\frac{1}{C} = -\frac{1}{-0.066663} = \mathbf{15.001\ mm}$$
Back focal distance (last surface → F’)
$$BFD = -\frac{A}{C} = -\frac{0.820630}{-0.066663} = \mathbf{12.310\ mm}$$
Front focal distance (first surface → F)
$$FFD = -\frac{D}{C} = -\frac{0.906960}{-0.066663} = \mathbf{13.605\ mm}$$
Image-side principal plane H’ (distance from last surface)
$$x_{H’} = \frac{1-A}{C} = \frac{1-0.820630}{-0.066663} = \frac{0.179370}{-0.066663} = \mathbf{-2.691\ mm}$$
Object-side principal plane H (distance from first surface)
$$x_H = \frac{D-1}{C} = \frac{0.906960-1}{-0.066663} = \frac{-0.093040}{-0.066663} = \mathbf{+1.396\ mm}$$
Foundations for Aberration Calculations
This chapter provides the shared paraxial ray-trace data used by the aberration calculations for spherical aberration, coma, astigmatism and distortion.
1. Entrance Pupil Radius
The f-stop is located directly at surface 1 (nothing in front of it) → entrance pupil = stop itself.
$$y_{marg} = \frac{f_{eff}}{2 \cdot N} = \frac{15.000000}{2 \cdot 3.5} = \frac{15.000000}{7} = \mathbf{2.142857\ mm}$$
2. Field Angle (from 8×11 mm format)
Step 1: \(\text{diagonal} = \sqrt{8^2 + 11^2} = \sqrt{64+121} = \sqrt{185} = \mathbf{13.601471\ mm}\)
Step 2: \(\text{half-diagonal } y’ = 13.601471 / 2 = \mathbf{6.800735\ mm}\)
Step 3: \(\tan(\omega) = y’/f_{eff} = 6.800735 / 15.000000 = \mathbf{0.453382}\)
Step 4: \(\omega = \arctan(0.453382) = \mathbf{0.425663\ rad = 24.389^\circ}\)
3. Starting Vectors [y, u] (u = real angle, radians)
- Marginal ray: \([y, u] = [2.142857\,;\, 0]\) (parallel to axis, object at infinity)
- Chief ray: \([y, u] = [0\,;\, 0.425663]\) (through stop center, at field angle \(\omega\))
4. Paraxial Trace — Marginal Ray
Start: \([y, u] = [2.142857\,;\, 0]\)
R1: [[1,0],[-0.088358, 0.620501]]
\(y’ = 1 \cdot 2.142857 = \mathbf{2.142857}\)
\(u’ = -0.088358 \cdot 2.142857 + 0.620501 \cdot 0 = \mathbf{-0.189339}\)
T1 (d=0.8): [[1,0.8],[0,1]]
\(y’ = 2.142857 + 0.8 \cdot (-0.189339) = 2.142857 – 0.151471 = \mathbf{1.991386}\)
\(u’ = \mathbf{-0.189339}\)
R2: [[1,0],[0.029124, 1.611600]]
\(y’ = \mathbf{1.991386}\)
\(u’ = 0.029124 \cdot 1.991386 + 1.611600 \cdot (-0.189339) = 0.058008 – 0.305150 = \mathbf{-0.247142}\)
T2 (d=1.126477): [[1,1.126477],[0,1]]
\(y’ = 1.991386 + 1.126477 \cdot (-0.247142) = 1.991386 – 0.278400 = \mathbf{1.712986}\)
\(u’ = \mathbf{-0.247142}\)
R3: [[1,0],[0.029834, 0.616219]]
\(y’ = \mathbf{1.712986}\)
\(u’ = 0.029834 \cdot 1.712986 + 0.616219 \cdot (-0.247142) = 0.051108 – 0.152329 = \mathbf{-0.101221}\) (precise value: -0.101189)
T3 (d=0.25): [[1,0.25],[0,1]]
\(y’ = 1.712986 + 0.25 \cdot (-0.101189) = 1.712986 – 0.025297 = \mathbf{1.687689}\)
\(u’ = \mathbf{-0.101189}\)
R4: [[1,0],[0.148286, 1.622800]]
\(y’ = \mathbf{1.687689}\)
\(u’ = 0.148286 \cdot 1.687689 + 1.622800 \cdot (-0.101189) = 0.250310 – 0.164221 = \mathbf{0.086089}\) (precise value: 0.086051)
T4 (d=0.952009): [[1,0.952009],[0,1]]
\(y’ = 1.687689 + 0.952009 \cdot 0.086051 = 1.687689 + 0.081924 = \mathbf{1.769613}\) (precise value: 1.769610)
\(u’ = \mathbf{0.086051}\)
R5: [[1,0],[-0.036455, 0.620501]]
\(y’ = \mathbf{1.769610}\)
\(u’ = -0.036455 \cdot 1.769610 + 0.620501 \cdot 0.086051 = -0.064525 + 0.053396 = \mathbf{-0.011129}\) (precise value: -0.011117)
T5 (d=1.0): [[1,1],[0,1]]
\(y’ = 1.769610 + 1 \cdot (-0.011117) = \mathbf{1.758493}\)
\(u’ = \mathbf{-0.011117}\)
R6: [[1,0],[-0.071050, 1.611600]]
\(y’ = \mathbf{1.758493}\)
\(u’ = -0.071050 \cdot 1.758493 + 1.611600 \cdot (-0.011117) = -0.124941 – 0.017916 = \mathbf{-0.142857}\)
Final marginal ray: \(y’ = 1.758493\text{ mm}\), \(u’ = -0.142857\text{ rad}\)
5. Paraxial Trace — Chief Ray
Start: \([y, u] = [0\,;\, 0.425663]\)
R1
\(y’ = 0\)
\(u’ = 0.620501 \cdot 0.425663 = \mathbf{0.264125}\)
T1 (d=0.8)
\(y’ = 0 + 0.8 \cdot 0.264125 = \mathbf{0.211300}\)
\(u’ = \mathbf{0.264125}\)
R2
\(y’ = \mathbf{0.211300}\)
\(u’ = 0.029124 \cdot 0.211300 + 1.611600 \cdot 0.264125 = 0.006154 + 0.425663 = \mathbf{0.431817}\)
T2 (d=1.126477)
\(y’ = 0.211300 + 1.126477 \cdot 0.431817 = 0.211300 + 0.486432 = \mathbf{0.697732}\) (precise: 0.697731)
\(u’ = \mathbf{0.431817}\)
R3
\(y’ = \mathbf{0.697731}\)
\(u’ = 0.029834 \cdot 0.697731 + 0.616219 \cdot 0.431817 = 0.020817 + 0.266093 = \mathbf{0.286910}\)
T3 (d=0.25)
\(y’ = 0.697731 + 0.25 \cdot 0.286910 = 0.697731 + 0.071728 = \mathbf{0.769459}\)
\(u’ = \mathbf{0.286910}\)
R4
\(y’ = \mathbf{0.769459}\)
\(u’ = 0.148286 \cdot 0.769459 + 1.622800 \cdot 0.286910 = 0.114104 + 0.465594 = \mathbf{0.579698}\) (precise: 0.579697)
T4 (d=0.952009)
\(y’ = 0.769459 + 0.952009 \cdot 0.579697 = 0.769459 + 0.551875 = \mathbf{1.321334}\) (precise: 1.321335)
\(u’ = \mathbf{0.579697}\)
R5
\(y’ = \mathbf{1.321335}\)
\(u’ = -0.036455 \cdot 1.321335 + 0.620501 \cdot 0.579697 = -0.048175 + 0.359708 = \mathbf{0.311533}\)
T5 (d=1.0)
\(y’ = 1.321335 + 1 \cdot 0.311533 = \mathbf{1.632868}\)
\(u’ = \mathbf{0.311533}\)
R6
\(y’ = \mathbf{1.632868}\)
\(u’ = -0.071050 \cdot 1.632868 + 1.611600 \cdot 0.311533 = -0.116015 + 0.502066 = \mathbf{0.386051}\)
Final chief ray: \(y’ = 1.632868\text{ mm}\), \(u’ = 0.386051\text{ rad}\)
6. Invariants Table
At each surface:
\(A = n \cdot (y_m \cdot c + u_{before,m})\)
\(\bar{A} = n \cdot (y_c \cdot c + u_{before,c})\)
\(H = n \cdot (u_{before,c} \cdot y_m – u_{before,m} \cdot y_c)\)
\(\Delta(u/n) = u_{after,m}/n’ – u_{before,m}/n\)
\(\Delta(1/n) = 1/n’ – 1/n\)
| Surf | R | c=1/R | y_m | u_before,m | u_after,m | y_c | u_before,c | u_after,c | A | Ā | H | Δ(u/n) | Δ(1/n) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 4.295000 | 0.232829 | 2.142857 | 0.000000 | -0.189339 | 0.000000 | 0.425663 | 0.264125 | 0.498919 | 0.425663 | 0.912135 | -0.117485 | -0.379499 |
| 2 | 21.000000 | 0.047619 | 1.991386 | -0.189339 | -0.247142 | 0.211300 | 0.264125 | 0.431817 | -0.152314 | 0.441879 | 0.912135 | -0.129657 | 0.379499 |
| 3 | -12.864000 | -0.077736 | 1.712986 | -0.247142 | -0.101189 | 0.697731 | 0.431817 | 0.286910 | -0.380303 | 0.377578 | 0.912135 | 0.184788 | -0.383781 |
| 4 | 4.200000 | 0.238095 | 1.687689 | -0.101189 | 0.086051 | 0.769459 | 0.286910 | 0.579697 | 0.487881 | 0.762901 | 0.912135 | 0.148405 | 0.383781 |
| 5 | 10.410000 | 0.096061 | 1.769610 | 0.086051 | -0.011117 | 1.321335 | 0.579697 | 0.311533 | 0.256042 | 0.706626 | 0.912135 | -0.092949 | -0.379499 |
| 6 | -8.608000 | -0.116171 | 1.758493 | -0.011117 | -0.142857 | 1.632868 | 0.311533 | 0.386051 | -0.347143 | 0.196359 | 0.912135 | -0.135959 | 0.379499 |
Check: H is identical at every surface (0.912135) — confirms correct computation, since H is a system-wide invariant.
This table is the shared basis for all four following aberration calculations.
Spherical Aberration (Seidel Sum_I)
Uses the marginal/chief ray trace and invariants table from “Foundations for Aberration Calculations” ), \(d_4 = 0.952009\text{ mm}\), \(f_{eff} = 15.000\text{ mm}\)).
1. Formula
For each surface k:
$$\text{contribution}_k = A_k^2 \cdot y_k \cdot \Delta(u/n)_k$$
$$S_I = -\sum \text{contribution}_k$$
where \(A = n \cdot (y \cdot c + u_{before})\) is the marginal-ray refraction invariant, \(y\) is the marginal ray height at the surface, and \(\Delta(u/n) = u_{after}/n’ – u_{before}/n\) (marginal ray).
2. Input Values (from foundations table)
| Surf | A | y_m | Δ(u/n) |
|---|---|---|---|
| 1 | 0.498919 | 2.142857 | -0.117485 |
| 2 | -0.152314 | 1.991386 | -0.129657 |
| 3 | -0.380303 | 1.712986 | 0.184788 |
| 4 | 0.487881 | 1.687689 | 0.148405 |
| 5 | 0.256042 | 1.769610 | -0.092949 |
| 6 | -0.347143 | 1.758493 | -0.135959 |
3. Step-by-Step Calculation per Surface
Surface 1
Step 1: \(A^2 = 0.498919^2 = \mathbf{0.248920}\)
Step 2: \(A^2 \cdot y = 0.248920 \cdot 2.142857 = \mathbf{0.533400}\)
Step 3: \(\text{contribution}_1 = 0.533400 \cdot (-0.117485) = \mathbf{-0.062667}\)
Surface 2
Step 1: \(A^2 = (-0.152314)^2 = \mathbf{0.023200}\)
Step 2: \(A^2 \cdot y = 0.023200 \cdot 1.991386 = \mathbf{0.046200}\)
Step 3: \(\text{contribution}_2 = 0.046200 \cdot (-0.129657) = \mathbf{-0.005990}\)
Surface 3
Step 1: \(A^2 = (-0.380303)^2 = \mathbf{0.144630}\)
Step 2: \(A^2 \cdot y = 0.144630 \cdot 1.712986 = \mathbf{0.247764}\)
Step 3: \(\text{contribution}_3 = 0.247764 \cdot 0.184788 = \mathbf{0.045782}\)
Surface 4
Step 1: \(A^2 = 0.487881^2 = \mathbf{0.238027}\)
Step 2: \(A^2 \cdot y = 0.238027 \cdot 1.687689 = \mathbf{0.401751}\)
Step 3: \(\text{contribution}_4 = 0.401751 \cdot 0.148405 = \mathbf{0.059618}\)
Surface 5
Step 1: \(A^2 = 0.256042^2 = \mathbf{0.065558}\)
Step 2: \(A^2 \cdot y = 0.065558 \cdot 1.769610 = \mathbf{0.116017}\)
Step 3: \(\text{contribution}_5 = 0.116017 \cdot (-0.092949) = \mathbf{-0.010785}\)
Surface 6
Step 1: \(A^2 = (-0.347143)^2 = \mathbf{0.120508}\)
Step 2: \(A^2 \cdot y = 0.120508 \cdot 1.758493 = \mathbf{0.211906}\)
Step 3: \(\text{contribution}_6 = 0.211906 \cdot (-0.135959) = \mathbf{-0.028812}\)
4. Summation
| Surface | contribution |
|---|---|
| 1 | -0.062667 |
| 2 | -0.005990 |
| 3 | 0.045782 |
| 4 | 0.059618 |
| 5 | -0.010785 |
| 6 | -0.028812 |
$$\sum \text{contribution} = \mathbf{-0.002854}$$
$$S_I = -\sum \text{contribution} = -(-0.002854) = \mathbf{+0.002854}$$
(High-precision value: \(S_I = 0.0028531\))
5. Interpretation
\(S_I\) is the third-order (Seidel) spherical aberration coefficient. Its small positive value indicates a small residual (undercorrected-direction) spherical aberration at full aperture (f/3.5) — consistent with a reasonably well-corrected triplet design; the individual surface contributions (ranging roughly ±0.06) show substantial internal cancellation between surfaces, which is typical of a balanced lens design.
Coma (Seidel Sum S_II)
Uses the marginal/chief ray trace and invariants table from “Foundations for Aberration Calculations” ( \(d_2 = 1.126477\text{ mm}\), \(d_4 = 0.952009\text{ mm}\), \(f_{eff} = 15.000\text{ mm}\)).
1. Formula
For each surface k:
$$\text{contribution}_k = \bar{A}_k \cdot A_k \cdot y_k \cdot \Delta(u/n)_k$$
$$S_{II} = -\sum \text{contribution}_k$$
where \(A\) is the marginal-ray invariant, \(\bar{A}\) the chief-ray invariant, \(y\) the marginal ray height, \(\Delta(u/n)\) the marginal ray’s reduced-angle change across the surface.
2. Input Values (from foundations table)
| Surf | A | Ā | y_m | Δ(u/n) |
|---|---|---|---|---|
| 1 | 0.498919 | 0.425663 | 2.142857 | -0.117485 |
| 2 | -0.152314 | 0.441879 | 1.991386 | -0.129657 |
| 3 | -0.380303 | 0.377578 | 1.712986 | 0.184788 |
| 4 | 0.487881 | 0.762901 | 1.687689 | 0.148405 |
| 5 | 0.256042 | 0.706626 | 1.769610 | -0.092949 |
| 6 | -0.347143 | 0.196359 | 1.758493 | -0.135959 |
3. Step-by-Step Calculation per Surface
Surface 1
Step 1: \(\bar{A} \cdot A = 0.425663 \cdot 0.498919 = \mathbf{0.212345}\)
Step 2: \(\bar{A} \cdot A \cdot y = 0.212345 \cdot 2.142857 = \mathbf{0.455024}\)
Step 3: \(\text{contribution}_1 = 0.455024 \cdot (-0.117485) = \mathbf{-0.053465}\)
Surface 2
Step 1: \(\bar{A} \cdot A = 0.441879 \cdot (-0.152314) = \mathbf{-0.067311}\)
Step 2: \(\bar{A} \cdot A \cdot y = -0.067311 \cdot 1.991386 = \mathbf{-0.134058}\)
Step 3: \(\text{contribution}_2 = -0.134058 \cdot (-0.129657) = \mathbf{0.017383}\)
Surface 3
Step 1: \(\bar{A} \cdot A = 0.377578 \cdot (-0.380303) = \mathbf{-0.143622}\)
Step 2: \(\bar{A} \cdot A \cdot y = -0.143622 \cdot 1.712986 = \mathbf{-0.246053}\)
Step 3: \(\text{contribution}_3 = -0.246053 \cdot 0.184788 = \mathbf{-0.045463}\)
Surface 4
Step 1: \(\bar{A} \cdot A = 0.762901 \cdot 0.487881 = \mathbf{0.372207}\)
Step 2: \(\bar{A} \cdot A \cdot y = 0.372207 \cdot 1.687689 = \mathbf{0.628210}\)
Step 3: \(\text{contribution}_4 = 0.628210 \cdot 0.148405 = \mathbf{0.093224}\)
Surface 5
Step 1: \(\bar{A} \cdot A = 0.706626 \cdot 0.256042 = \mathbf{0.180957}\)
Step 2: \(\bar{A} \cdot A \cdot y = 0.180957 \cdot 1.769610 = \mathbf{0.320205}\)
Step 3: \(\text{contribution}_5 = 0.320205 \cdot (-0.092949) = \mathbf{-0.029769}\)
Surface 6
Step 1: \(\bar{A} \cdot A = 0.196359 \cdot (-0.347143) = \mathbf{-0.068171}\)
Step 2: \(\bar{A} \cdot A \cdot y = -0.068171 \cdot 1.758493 = \mathbf{-0.119888}\)
Step 3: \(\text{contribution}_6 = -0.119888 \cdot (-0.135959) = \mathbf{0.016301}\)
4. Summation
| Surface | contribution |
|---|---|
| 1 | -0.053465 |
| 2 | 0.017383 |
| 3 | -0.045463 |
| 4 | 0.093224 |
| 5 | -0.029769 |
| 6 | 0.016301 |
$$\sum \text{contribution} = \mathbf{-0.001789} \quad \text{(high-precision: -0.0017796)}$$
$$S_{II} = -\sum \text{contribution} = -(-0.0017796) = \mathbf{+0.001780}$$
5. Interpretation
\(S_{II}\) is the third-order coma coefficient. Its small magnitude relative to the individual surface contributions (up to ±0.09) again shows strong internal cancellation — a hallmark of a well-corrected triplet, where coma from the front and rear groups largely offsets that of the middle (negative) element.
Astigmatism (Seidel Sum S_III)
Uses the marginal/chief ray trace and invariants table from “Foundations for Aberration Calculations” (corrected \(d_2 = 1.126477\text{ mm}\), \(d_4 = 0.952009\text{ mm}\), \(f_{eff} = 15.000\text{ mm}\)).
1. Formula
For each surface k:
$$\text{contribution}_k = \bar{A}_k^2 \cdot y_k \cdot \Delta(u/n)_k$$
$$S_{III} = -\sum \text{contribution}_k$$
where \(\bar{A}\) is the chief-ray refraction invariant, \(y\) the marginal ray height, \(\Delta(u/n)\) the marginal ray’s reduced-angle change across the surface.
2. Input Values (from foundations table)
| Surf | Ā | y_m | Δ(u/n) |
|---|---|---|---|
| 1 | 0.425663 | 2.142857 | -0.117485 |
| 2 | 0.441879 | 1.991386 | -0.129657 |
| 3 | 0.377578 | 1.712986 | 0.184788 |
| 4 | 0.762901 | 1.687689 | 0.148405 |
| 5 | 0.706626 | 1.769610 | -0.092949 |
| 6 | 0.196359 | 1.758493 | -0.135959 |
3. Step-by-Step Calculation per Surface
Surface 1
Step 1: \(\bar{A}^2 = 0.425663^2 = \mathbf{0.181189}\)
Step 2: \(\bar{A}^2 \cdot y = 0.181189 \cdot 2.142857 = \mathbf{0.388262}\)
Step 3: \(\text{contribution}_1 = 0.388262 \cdot (-0.117485) = \mathbf{-0.045615}\)
Surface 2
Step 1: \(\bar{A}^2 = 0.441879^2 = \mathbf{0.195257}\)
Step 2: \(\bar{A}^2 \cdot y = 0.195257 \cdot 1.991386 = \mathbf{0.388887}\)
Step 3: \(\text{contribution}_2 = 0.388887 \cdot (-0.129657) = \mathbf{-0.050415}\)
Surface 3
Step 1: \(\bar{A}^2 = 0.377578^2 = \mathbf{0.142605}\)
Step 2: \(\bar{A}^2 \cdot y = 0.142605 \cdot 1.712986 = \mathbf{0.244304}\)
Step 3: \(\text{contribution}_3 = 0.244304 \cdot 0.184788 = \mathbf{0.045143}\)
Surface 4
Step 1: \(\bar{A}^2 = 0.762901^2 = \mathbf{0.582017}\)
Step 2: \(\bar{A}^2 \cdot y = 0.582017 \cdot 1.687689 = \mathbf{0.982457}\)
Step 3: \(\text{contribution}_4 = 0.982457 \cdot 0.148405 = \mathbf{0.145811}\)
Surface 5
Step 1: \(\bar{A}^2 = 0.706626^2 = \mathbf{0.499320}\)
Step 2: \(\bar{A}^2 \cdot y = 0.499320 \cdot 1.769610 = \mathbf{0.883510}\)
Step 3: \(\text{contribution}_5 = 0.883510 \cdot (-0.092949) = \mathbf{-0.082131}\)
Surface 6
Step 1: \(\bar{A}^2 = 0.196359^2 = \mathbf{0.038557}\)
Step 2: \(\bar{A}^2 \cdot y = 0.038557 \cdot 1.758493 = \mathbf{0.067814}\)
Step 3: \(\text{contribution}_6 = 0.067814 \cdot (-0.135959) = \mathbf{-0.009221}\)
4. Summation
| Surface | contribution |
|---|---|
| 1 | -0.045615 |
| 2 | -0.050415 |
| 3 | 0.045143 |
| 4 | 0.145811 |
| 5 | -0.082131 |
| 6 | -0.009221 |
$$\sum \text{contribution} = \mathbf{0.003572} \quad \text{(high-precision: 0.0035229)}$$
$$S_{III} = -\sum \text{contribution} = -0.0035229 = \mathbf{-0.003523}$$
5. Interpretation
\(S_{III}\) is the third-order astigmatism coefficient. The small negative value indicates a slight residual astigmatism at the edge of the 8×11 mm format (field angle \(\omega \approx 24.4^\circ\)); combined with the Petzval curvature (\(S_{IV}\), see separate document) it determines the actual sagittal and tangential field curves of the system.
Petzval field curvature (Seidel Sum S_IV)
0. Input Data
| Surface | R (mm) | n (before) | n’ (after) |
|---|---|---|---|
| 1 | 4.295000 | 1.000000 | 1.611600 |
| 2 | 21.000000 | 1.611600 | 1.000000 |
| 3 | -12.864000 | 1.000000 | 1.622800 |
| 4 | 4.200000 | 1.622800 | 1.000000 |
| 5 | 10.410000 | 1.000000 | 1.611600 |
| 6 | -8.608000 | 1.611600 | 1.000000 |
1. Formula
For each surface k:
$$P_k = \frac{n’ – n}{n \cdot n’ \cdot R}$$
Total Petzval sum: $$P = P_1 + P_2 + P_3 + P_4 + P_5 + P_6$$
Petzval radius: $$r_P = \frac{1}{P}$$
2. Step-by-Step Calculation
Surface 1 (R=4.295; n=1; n’=1.6116)
Step 1: \(\Delta n = n’-n = 1.6116-1 = \mathbf{0.6116}\)
Step 2: \(n \cdot n’ = 1 \cdot 1.6116 = \mathbf{1.6116}\)
Step 3: \(n \cdot n’ \cdot R = 1.6116 \cdot 4.295 = \mathbf{6.921822}\)
Step 4: \(P_1 = 0.6116 / 6.921822 = \mathbf{0.088358}\)
Surface 2 (R=21.0; n=1.6116; n’=1)
Step 1: \(\Delta n = 1-1.6116 = \mathbf{-0.6116}\)
Step 2: \(n \cdot n’ = 1.6116 \cdot 1 = \mathbf{1.6116}\)
Step 3: \(n \cdot n’ \cdot R = 1.6116 \cdot 21.0 = \mathbf{33.8436}\)
Step 4: \(P_2 = -0.6116 / 33.8436 = \mathbf{-0.018071}\)
Surface 3 (R=-12.864; n=1; n’=1.6228)
Step 1: \(\Delta n = 1.6228-1 = \mathbf{0.6228}\)
Step 2: \(n \cdot n’ = 1 \cdot 1.6228 = \mathbf{1.6228}\)
Step 3: \(n \cdot n’ \cdot R = 1.6228 \cdot (-12.864) = \mathbf{-20.875699}\)
Step 4: \(P_3 = 0.6228 / (-20.875699) = \mathbf{-0.029834}\)
Surface 4 (R=4.2; n=1.6228; n’=1)
Step 1: \(\Delta n = 1-1.6228 = \mathbf{-0.6228}\)
Step 2: \(n \cdot n’ = 1.6228 \cdot 1 = \mathbf{1.6228}\)
Step 3: \(n \cdot n’ \cdot R = 1.6228 \cdot 4.2 = \mathbf{6.81576}\)
Step 4: \(P_4 = -0.6228 / 6.81576 = \mathbf{-0.091376}\)
Surface 5 (R=10.41; n=1; n’=1.6116)
Step 1: \(\Delta n = 1.6116-1 = \mathbf{0.6116}\)
Step 2: \(n \cdot n’ = 1 \cdot 1.6116 = \mathbf{1.6116}\)
Step 3: \(n \cdot n’ \cdot R = 1.6116 \cdot 10.41 = \mathbf{16.776756}\)
Step 4: \(P_5 = 0.6116 / 16.776756 = \mathbf{0.036455}\)
Surface 6 (R=-8.6080; n=1.6116; n’=1)
Step 1: \(\Delta n = 1-1.6116 = \mathbf{-0.6116}\)
Step 2: \(n \cdot n’ = 1.6116 \cdot 1 = \mathbf{1.6116}\)
Step 3: \(n \cdot n’ \cdot R = 1.6116 \cdot (-8.6080) = \mathbf{-13.872653}\)
Step 4: \(P_6 = -0.6116 / (-13.872653) = \mathbf{0.044087}\)
3. Summation
$$\mathbf{P = P_1+P_2+P_3+P_4+P_5+P_6=0.029619\ mm^{-1}}$$
4. Result
$$r_P = \frac{1}{P} = \frac{1}{0.029619} = \mathbf{33.763\ mm}$$
Compared with the corrected effective focal length \(f_{eff} = 15.000\text{ mm}\), the ratio \(r_P/f_{eff} \approx 2.25\) — again a moderate field curvature, typical of a well-corrected triplet.
Distortion (Seidel Sum S_V)
Uses the marginal/chief ray trace and invariants table from “Foundations for Aberration Calculations” (\(d_2 = 1.126477\text{ mm}\), \(d_4 = 0.952009\text{ mm}\), \(f_{eff} = 15.000\text{ mm}\)). Also requires the Petzval-related Lagrange invariant \(H\) and \(\Delta(1/n)\), and the curvature \(c = 1/R\).
1. Formula
For each surface k:
$$\text{contribution}_k = \left(\frac{\bar{A}_k^3}{A_k}\right) \cdot y_k \cdot \Delta(u/n)_k + \left(\frac{\bar{A}_k}{A_k}\right) \cdot H^2 \cdot c_k \cdot \Delta(1/n)_k$$
$$S_V = -\sum \text{contribution}_k$$
where \(A\), \(\bar{A}\) are the marginal/chief-ray invariants, \(H\) the (system-constant) Lagrange invariant, \(c\) the surface curvature, \(\Delta(u/n)\) the marginal ray’s angle change, \(\Delta(1/n)\) the index-reciprocal change.
2. Input Values (from foundations table)
| Surf | A | Ā | y_m | Δ(u/n) | H | c | Δ(1/n) |
|---|---|---|---|---|---|---|---|
| 1 | 0.498919 | 0.425663 | 2.142857 | -0.117485 | 0.912135 | 0.232829 | -0.379499 |
| 2 | -0.152314 | 0.441879 | 1.991386 | -0.129657 | 0.912135 | 0.047619 | 0.379499 |
| 3 | -0.380303 | 0.377578 | 1.712986 | 0.184788 | 0.912135 | -0.077736 | -0.383781 |
| 4 | 0.487881 | 0.762901 | 1.687689 | 0.148405 | 0.912135 | 0.238095 | 0.383781 |
| 5 | 0.256042 | 0.706626 | 1.769610 | -0.092949 | 0.912135 | 0.096061 | -0.379499 |
| 6 | -0.347143 | 0.196359 | 1.758493 | -0.135959 | 0.912135 | -0.116171 | 0.379499 |
$$H^2 = 0.912135^2 = \mathbf{0.831990} \quad \text{(constant for all surfaces)}$$
3. Step-by-Step Calculation per Surface
Surface 1
Term A (coma-like part):
Step 1: \(\bar{A}^3 = 0.425663^3 = 0.425663 \cdot 0.425663 \cdot 0.425663 = \mathbf{0.077120}\)
Step 2: \(\bar{A}^3/A = 0.077120 / 0.498919 = \mathbf{0.154591}\)
Step 3: \((\bar{A}^3/A) \cdot y = 0.154591 \cdot 2.142857 = \mathbf{0.331267}\)
Step 4: \(\text{term}_A = 0.331267 \cdot (-0.117485) = \mathbf{-0.038919}\)
Term B (Petzval-related part):
Step 5: \(\bar{A}/A = 0.425663 / 0.498919 = \mathbf{0.853176}\)
Step 6: \(H^2 \cdot c = 0.831990 \cdot 0.232829 = \mathbf{0.193708}\)
Step 7: \(H^2 \cdot c \cdot \Delta(1/n) = 0.193708 \cdot (-0.379499) = \mathbf{-0.073512}\)
Step 8: \(\text{term}_B = 0.853176 \cdot (-0.073512) = \mathbf{-0.062722}\)
\(\text{contribution}_1 = \text{term}_A + \text{term}_B = -0.038919 + (-0.062722) = \mathbf{-0.101641}\)
Surface 2
Step 1: \(\bar{A}^3 = 0.441879^3 = \mathbf{0.086265}\)
Step 2: \(\bar{A}^3/A = 0.086265 / (-0.152314) = \mathbf{-0.566368}\)
Step 3: \((\bar{A}^3/A) \cdot y = -0.566368 \cdot 1.991386 = \mathbf{-1.127977}\)
Step 4: \(\text{term}_A = -1.127977 \cdot (-0.129657) = \mathbf{0.146259}\)
Step 5: \(\bar{A}/A = 0.441879 / (-0.152314) = \mathbf{-2.900909}\)
Step 6: \(H^2 \cdot c = 0.831990 \cdot 0.047619 = \mathbf{0.039619}\)
Step 7: \(H^2 \cdot c \cdot \Delta(1/n) = 0.039619 \cdot 0.379499 = \mathbf{0.015035}\)
Step 8: \(\text{term}_B = -2.900909 \cdot 0.015035 = \mathbf{-0.043621}\)
\(\text{contribution}_2 = 0.146259 + (-0.043621) = \mathbf{0.102638}\)
Surface 3
Step 1: \(\bar{A}^3 = 0.377578^3 = \mathbf{0.053832}\)
Step 2: \(\bar{A}^3/A = 0.053832 / (-0.380303) = \mathbf{-0.141550}\)
Step 3: \((\bar{A}^3/A) \cdot y = -0.141550 \cdot 1.712986 = \mathbf{-0.242514}\)
Step 4: \(\text{term}_A = -0.242514 \cdot 0.184788 = \mathbf{-0.044811}\)
Step 5: \(\bar{A}/A = 0.377578 / (-0.380303) = \mathbf{-0.992836}\)
Step 6: \(H^2 \cdot c = 0.831990 \cdot (-0.077736) = \mathbf{-0.064676}\)
Step 7: \(H^2 \cdot c \cdot \Delta(1/n) = -0.064676 \cdot (-0.383781) = \mathbf{0.024820}\)
Step 8: \(\text{term}_B = -0.992836 \cdot 0.024820 = \mathbf{-0.024642}\)
\(\text{contribution}_3 = -0.044811 + (-0.024642) = \mathbf{-0.069453}\)
Surface 4
Step 1: \(\bar{A}^3 = 0.762901^3 = \mathbf{0.443938}\)
Step 2: \(\bar{A}^3/A = 0.443938 / 0.487881 = \mathbf{0.909910}\)
Step 3: \((\bar{A}^3/A) \cdot y = 0.909910 \cdot 1.687689 = \mathbf{1.535722}\)
Step 4: \(\text{term}_A = 1.535722 \cdot 0.148405 = \mathbf{0.227916}\)
Step 5: \(\bar{A}/A = 0.762901 / 0.487881 = \mathbf{1.563723}\)
Step 6: \(H^2 \cdot c = 0.831990 \cdot 0.238095 = \mathbf{0.198093}\)
Step 7: \(H^2 \cdot c \cdot \Delta(1/n) = 0.198093 \cdot 0.383781 = \mathbf{0.076031}\)
Step 8: \(\text{term}_B = 1.563723 \cdot 0.076031 = \mathbf{0.118903}\)
\(\text{contribution}_4 = 0.227916 + 0.118903 = \mathbf{0.346819}\)
Surface 5
Step 1: \(\bar{A}^3 = 0.706626^3 = \mathbf{0.352892}\)
Step 2: \(\bar{A}^3/A = 0.352892 / 0.256042 = \mathbf{1.378254}\)
Step 3: \((\bar{A}^3/A) \cdot y = 1.378254 \cdot 1.769610 = \mathbf{2.439116}\)
Step 4: \(\text{term}_A = 2.439116 \cdot (-0.092949) = \mathbf{-0.226755}\)
Step 5: \(\bar{A}/A = 0.706626 / 0.256042 = \mathbf{2.759723}\)
Step 6: \(H^2 \cdot c = 0.831990 \cdot 0.096061 = \mathbf{0.079922}\)
Step 7: \(H^2 \cdot c \cdot \Delta(1/n) = 0.079922 \cdot (-0.379499) = \mathbf{-0.030330}\)
Step 8: \(\text{term}_B = 2.759723 \cdot (-0.030330) = \mathbf{-0.083691}\)
\(\text{contribution}_5 = -0.226755 + (-0.083691) = \mathbf{-0.310446}\)
Surface 6
Step 1: \(\bar{A}^3 = 0.196359^3 = \mathbf{0.007572}\)
Step 2: \(\bar{A}^3/A = 0.007572 / (-0.347143) = \mathbf{-0.021811}\)
Step 3: \((\bar{A}^3/A) \cdot y = -0.021811 \cdot 1.758493 = \mathbf{-0.038354}\)
Step 4: \(\text{term}_A = -0.038354 \cdot (-0.135959) = \mathbf{0.005215}\)
Step 5: \(\bar{A}/A = 0.196359 / (-0.347143) = \mathbf{-0.565744}\)
Step 6: \(H^2 \cdot c = 0.831990 \cdot (-0.116171) = \mathbf{-0.096650}\)
Step 7: \(H^2 \cdot c \cdot \Delta(1/n) = -0.096650 \cdot 0.379499 = \mathbf{-0.036680}\)
Step 8: \(\text{term}_B = -0.565744 \cdot (-0.036680) = \mathbf{0.020753}\)
\(\text{contribution}_6 = 0.005215 + 0.020753 = \mathbf{0.025968}\)
4. Summation
| Surface | contribution |
|---|---|
| 1 | -0.101641 |
| 2 | 0.102638 |
| 3 | -0.069453 |
| 4 | 0.346819 |
| 5 | -0.310446 |
| 6 | 0.025968 |
$$\sum \text{contribution} = \mathbf{-0.006115} \quad \text{(high-precision: -0.0060251)}$$
$$S_V = -\sum \text{contribution} = -(-0.0060251) = \mathbf{+0.006025}$$
5. Interpretation
\(S_V\) is the third-order distortion coefficient. A positive value in this convention corresponds to pincushion-direction distortion at the edge of the 8×11 mm field; the magnitude (\(\approx 0.006\)) is small relative to the intermediate per-surface terms (up to \(\pm 0.35\)), again showing strong internal balancing typical of a well-designed triplet.
Optical-Historical Evaluation
1. Summary of Aberration Sums
Seidel Aberration Sums (3rd Order)
| Aberration Coefficient | Symbol | Value | Residual Trend |
|---|---|---|---|
| Spherical Aberration | SI | +0.00285 | Minor undercorrected residual |
| Coma | SII | +0.00178 | Extremely low residual coma |
| Astigmatism | SIII | -0.00352 | Slight negative astigmatism at field edge |
| Field Curvature (Petzval) | SIV | +0.02962 | Moderate Petzval curvature (rP ≈ 2.25 · f) |
| Distortion | SV | +0.00603 | Minor pincushion distortion |
2. Optical Evaluation of the 1939 Minostigmat Design
Internal Compensation & Balance
The breakdown of individual surface contributions illustrates the fundamental operating principle of the Cooke triplet. The outer crown elements introduce large positive and negative aberration contributions—for example, individual surface distortion terms (SV) reach peak values of +0.35 and -0.31. The central negative element cancels these deviations almost entirely.
The resulting net Seidel sums remain exceptionally close to zero (|SI|, |SII|, |SIII|, |SV| ≪ 0.01), demonstrating a remarkably well-balanced optical design.
Performance at Full Aperture (f/3.5)
- On-Axis Sharpness (SI = +0.00285): Spherical aberration is superbly corrected for a three-element lens operating at f/3.5. Stopping down slightly to f/4.5 or f/5.6 would drive axial performance straight to the diffraction limit.
- Off-Axis Field Performance (SIII, SIV, SV): Mapping a 49° diagonal field of view onto the 8 × 11 mm frame places noticeable demands on a 15 mm triplet. However, the subtle interplay between residual astigmatism (SIII = -0.0035) and Petzval curvature (rP ≈ 33.8 mm) yields a flat usable image plane across the frame, minimizing edge soft focus. Distortion (SV ≈ +0.006) remains visually imperceptible.
Manufacturing context and mechanical sensitivity
Utilizing optical glasses with refractive indices ranging from n ≈ 1.6116 to 1.6228 (standard high-index crown and flint glasses of the late 1930s), the design fully exploited the available degree of freedom in surface radii and air spaces.
Furthermore, the precise alignment of air gaps (d2 = 1.126 mm, d4 = 0.952 mm) underscores how sensitive short-focal-length triplets (15 mm) are to mechanical assembly tolerances. Spacing variations of merely a few hundredths of a millimeter noticeably alter both the back focal length and the overall state of aberration correction.
3. Empirical Verification: Photographic Resolution Test
Practical resolution measurements conducted using high-resolution ADOX CMS 20 II film fully confirm the theoretical calculations:
| Image Region | Measured Resolution | Film Stock | Theoretical Agreement |
|---|---|---|---|
| Center (Axial) | 203 LP/mm | ADOX CMS 20 II | A strong match to the calculated circle of least confusion |
| Corners (8 × 11 mm edge) | 114 LP/mm | ADOX CMS 20 II | An expected, substantial drop driven by Petzval curvature (~9× size) |
Center: 203 LP/mm — A Strong Match
From the calculated Seidel spherical aberration sum SI, the transverse ray aberration of the marginal ray at the paraxial focus can be estimated:
For third-order spherical aberration, the best achievable focus (circle of least confusion) lies not at the paraxial focus but roughly three-quarters of the way toward the marginal-ray focus, where the blur radius shrinks to about TSC / 4:
This gives a rough resolving-power estimate:
This matches the measured 203 LP/mm almost exactly. For such a coarse estimate, the agreement is striking — a strong indication that both the input data (radii, indices) and the corrected air gaps d2 / d4 used in this calculation are close to the lens’s true physical values.
Corners: 114 LP/mm — An Expected, Substantial Drop
At the edge of the field (ω ≈ 24.4°, corresponding to the corner of the 8 × 11 mm format), coma, astigmatism, and above all field curvature come into play. Comparing the magnitudes of the Seidel sums calculated earlier:
| Aberration | Value | Relative Size |
|---|---|---|
| SI (spherical) | 0.00285 | 1× |
| SII (coma) | 0.00178 | 0.6× |
| SIII (astigmatism) | -0.00352 | 1.2× |
| SIV (Petzval) | 0.02464 | ~9× |
| SV (distortion) | 0.00603 | 2.1× |
The calculated Petzval sum is by far the largest aberration coefficient — field curvature is by far the dominant factor degrading edge performance, exactly as expected for a simple triplet. This is consistent with the Petzval radius calculated earlier (rP ≈ 33.8 mm), only about 2.25× the focal length — a noticeably curved image shell falling on a flat film plane.
Combining the coma, astigmatism, and defocus contributions at the edge plausibly yields a resolving power well below the center — a drop to roughly 110–150 LP/mm is exactly what one would expect for a half-field angle this large (±24°) in an uncorrected triplet. The measured 114 LP/mm falls squarely within this predicted range.
Final conclusion
The optical performance of the 15 mm f/3.5 Minostigmat lens represents a remarkable engineering achievement for its time. Balancing an asymmetrical three-element Cooke triplet so effectively that both spherical aberration and coma approach diffraction-limited thresholds in the center is an extraordinary optical feat.
During its development, designer Hans Schulz deliberately prioritized maximum axial sharpness and high central contrast, ensuring that the tiny negatives could sustain significant enlargement. As a consequence, Petzval field curvature was accepted as the primary physical compromise inherent to the simple triplet layout, which accounts for the resolution drop from 203 LP/mm at the center to 114 LP/mm at the corners on flat film.
Ultimately, the lens delivers outstanding central sharpness and contrast, leaving edge fall-off to be driven almost entirely by field curvature rather than coma or centering defects, resulting in an exceptionally well-balanced correction profile for a 1930s subminiature optic.
The previous practical test on ADOX CMS 20 II proves that this theoretical performance translates directly into exceptional real-world resolving power across the entire 8 x 11 mm subminiature format.
The calculations presented in this article would take about 2 to 3 hours to perform using a calculator. If, as was the case in the 1930s, one had to rely solely on logarithm tables, it would likely take 12 to 18 hours of work. In addition, the matrix method was not yet known at that time, so calculating the focal lengths took considerably longer. Finally, it’s important to keep in mind that we have simply recalculated an existing triplet here. Designing such a lens system required many more such iterations.
More articles about VEF Minox Riga
- My story of who developed the Minostigmat
- My in-depth hands-on review of the Minostigmat lens
- Burkhard’s report on his complete restoration of a VEF Minox Riga
- My detailed introduction to VEF Minox Riga
