Chapter 18: How to Read Optical Software Outputs Now

At the beginning of this book, an MTF curve looked like a finished object.

A piece of software produced it. The curve went down with frequency. There were sagittal and tangential lines. There were field positions, wavelengths, and maybe a diffraction limit. It looked official.

Now we can read that curve differently.

It is no longer a mysterious judgment from the software.

It is the end of a chain:

lens prescription
→ surfaces and materials
→ ray tracing
→ optical path length
→ OPD
→ pupil function
→ PSF
→ OTF
→ MTF

And if optimization is involved, the chain becomes a loop:

prescription
→ analysis
→ merit function
→ optimizer changes variables
→ new prescription
→ analysis again

That is the main change this book wanted to make.

Not that you should stop using optical design software. Quite the opposite. Good optical design software is powerful precisely because it performs so many hard calculations reliably.

You no longer have to treat the output as a sealed verdict.

You can ask where it came from.

You can inspect its inputs.

You can check its assumptions.

You can compare it against the rest of the system.

That is what it means to read optical software outputs well.

1. The first habit: never read a plot alone

A plot is not self-explaining.

A spot diagram, ray fan, wavefront map, PSF, or MTF curve always belongs to a configuration.

Before interpreting any output, ask:

What optical system produced this?
What field point?
What wavelength or wavelength set?
What aperture?
What focus setting?
What sampling?
What normalization?
What coordinate convention?

This may feel slow at first. It is not slow. It is the shortest path to not fooling yourself.

A plot without its configuration is like a number without a unit.

It may look precise, but you do not yet know what it means.

For example, this statement is weak:

The MTF is 0.45.

This statement is better:

At 40 cycles/mm, for the edge field, green wavelength,
tangential direction, full aperture, and current focus setting,
the normalized MTF is 0.45.

It is longer because it is more honest.

Most optical misunderstandings begin when a short statement hides the configuration.

2. Start from the prescription

When you see a software output, do not start by admiring the plot.

Start by asking what prescription produced it.

A lens prescription contains the basic computed object:

surface radius
thickness
material
semi-aperture
conic or aspheric terms
stop position
image plane

This is not paperwork. It is the system.

Every later calculation depends on it.

A practical reading sequence is:

1. Prescription
2. Layout
3. Paraxial properties
4. Real ray analysis
5. Wavefront analysis
6. PSF / MTF
7. Optimization report

If the prescription is wrong, every later output is downstream of that error.

So the first check is plain:

Does this model represent the optical system I think it represents?

Look for simple mistakes:

• radius sign errors;
• wrong glass;
• thickness in the wrong unit;
• missing stop;
• wrong image distance;
• semi-aperture too small;
• surface order mistake;
• wavelength definition mismatch;
• fields entered in the wrong unit.

These are not glamorous errors. They are common.

Optical computation is full of advanced mathematics, but many bad results begin with ordinary bad input.

3. Then look at the layout

The layout is the first visual sanity check.

It tells you whether the prescription has become a plausible physical system.

Ask:

Do the elements appear in the right order?
Do rays pass through the aperture?
Does the stop appear where expected?
Do rays reach the image plane?
Are there obvious ray failures?
Does the scale look reasonable?

The layout does not prove image quality. A bad lens can have a perfectly reasonable layout.

But an obviously wrong layout makes later plots untrustworthy.

This is a useful rule:

Do not trust a beautiful MTF curve from a lens layout you have not checked.

The layout is the first defense against hidden modeling mistakes.

4. Use paraxial results as the skeleton check

Paraxial analysis is not old-fashioned decoration. It is the skeleton of the optical system.

Before reading detailed image-quality plots, check:

effective focal length
back focal length
f-number
numerical aperture
pupil diameter
magnification
chief ray behavior

These are first-order quantities. They tell you whether the system is roughly the lens you intended to build.

If you expected a 50 mm lens and the paraxial analysis gives 500 mm, do not continue interpreting the PSF. Something is wrong upstream.

If you expected f/4 and the computed aperture behaves like f/16, your diffraction scale and MTF cutoff will be different.

If the entrance pupil is not where you expect, field behavior and aperture sampling may surprise you.

The paraxial result is not the final truth. Real rays will depart from it.

But it gives the reference frame.

Paraxial analysis tells you the optical system’s first-order identity.
Real ray analysis tells you how reality departs from that identity.

Read them in that order.

5. Spot diagrams: read them as ray intercept statistics

A spot diagram is not a drawing of blur in the wave-optical sense.

It is a distribution of ray intercepts at an image plane.

It answers:

Where do sampled rays land?

When reading a spot diagram, ask:

Which field point is this?
Which wavelengths are included?
How were pupil rays sampled?
Is the spot shown relative to the chief ray, centroid, or ideal image point?
What is the scale bar?
Is the Airy disk shown?

The Airy disk comparison is especially useful.

If the geometric spot is much larger than the Airy disk, geometric aberration is probably important.

If the geometric spot is much smaller than the Airy disk, diffraction may dominate, and spot size alone may not tell the whole story.

Do not overread a spot diagram near the diffraction limit.

A spot diagram can be very useful, but it is still geometric.

It does not show diffraction rings.

It does not directly show contrast transfer.

It does not replace PSF or MTF.

A careful reading is:

This spot diagram shows the geometric landing pattern of sampled rays
for this field, wavelength set, aperture, and image plane.

That sentence keeps the plot in its proper place.

6. Ray fans: look for structure, not only size

A ray fan shows ray error as a function of pupil coordinate.

It answers:

How does ray error vary across the pupil?

This makes it more diagnostic than a spot diagram in many cases.

When reading a ray fan, ask:

Is the curve symmetric or asymmetric?
Does it show spherical-like behavior?
Does it show coma-like behavior?
Do tangential and sagittal fans differ?
Do different wavelengths separate?
Does the fan flatten after optimization?

A spot diagram may show a cloud. The ray fan can show why the cloud has that shape.

Use it as a structural diagnostic.

For example:

• a symmetric fan on-axis may suggest spherical aberration or defocus-like behavior;
• an asymmetric fan off-axis may suggest coma-like behavior;
• different tangential and sagittal behavior may point toward astigmatism;
• color-separated fans point toward chromatic effects.

The ray fan is not just another plot.

It is one of the best ways to see how real rays depart from the ideal reference.

7. Wavefront and OPD: ask what reference is being used

OPD is the bridge between ray optics and diffraction imaging.

It answers:

How far ahead or behind is each pupil point compared with a reference wavefront?

When reading an OPD or wavefront plot, ask:

What is the reference wavefront?
What wavelength is used?
Are units shown in length or waves?
Is piston removed?
Is tilt removed?
Is defocus removed?
What pupil coordinates are used?
Is the pupil vignetted or clipped?

These questions matter.

The same physical wavefront can look different depending on whether piston, tilt, or defocus has been removed. That does not mean the software is lying. It means wavefront plots are referenced quantities.

OPD in meters or micrometers is a physical path difference.

OPD in waves is normalized by wavelength:

$$W_\text{waves} = \frac{W}{\lambda}$$

So a wavefront error that is small in micrometers may still be large in waves at short wavelength.

This is why units matter so much.

A good reading habit is:

Before interpreting a wavefront map, identify the wavelength, reference, removed terms, and pupil definition.

Only then decide what the map says.

8. Pupil function: remember amplitude and phase

The pupil function combines aperture amplitude and wavefront phase:

$$P(x,y)=A(x,y)\exp\left(i\frac{2\pi W(x,y)}{\lambda}\right)$$

When a software tool computes PSF or MTF by diffraction, something equivalent to this representation is usually involved.

So when reading a diffraction result, ask:

What aperture mask was used?
Is there vignetting?
Is the amplitude uniform?
Is there apodization?
What OPD produced the phase?
Which wavelength was used?
What sign convention is used?

This matters because PSF changes do not only come from wavefront aberration.

They can also come from pupil amplitude:

• central obstruction;
• vignetting;
• aperture clipping;
• apodization;
• transmission variation;
• diffractive or phase elements.

A clear aberrated lens may have uniform amplitude but nonuniform phase.

An obstructed system may have flat phase but nonuniform amplitude.

Both affect the PSF.

Do not read every PSF feature as “aberration.” Some features are aperture structure.

9. PSF: read it as energy distribution

The PSF answers:

What image-plane intensity distribution does one ideal point produce?

When reading a PSF plot, ask:

Is the display linear or logarithmic?
Is the PSF normalized by energy or peak?
What is the image-plane coordinate unit?
What field and wavelength does it represent?
Is it monochromatic or polychromatic?
Is the central peak centered?
Is the plot cropped?
What is the sampling?

Linear and logarithmic PSF displays tell different visual stories.

A linear plot emphasizes the central energy concentration.

A log plot reveals weak rings, tails, and halos.

Neither is “the true one.” They are two displays of the same data.

Read both when possible.

Also distinguish energy normalization from peak normalization.

Energy normalization:

$$\sum \mathrm{PSF}=1$$

is useful for comparing how energy is distributed.

Peak normalization:

$$\max(\mathrm{PSF})=1$$

is useful for comparing shapes visually.

But peak-normalized PSFs cannot be used directly for Strehl comparison because the peak has already been forced to one.

A careful PSF reading is:

Where is the point-source energy going,
under this field, wavelength, aperture, focus, sampling, and normalization?

That is the question the PSF answers.

10. MTF: read it as a selected view of frequency response

MTF is the magnitude of the OTF:

$$\mathrm{OTF}=\mathcal{F}{\mathrm{PSF}}$$

$$\mathrm{MTF}=|\mathrm{OTF}|$$

It answers:

How strongly does the system transfer contrast at different spatial frequencies?

But a plotted MTF curve is usually not the whole 2D MTF.

It is a selected slice or set of slices.

When reading an MTF curve, ask:

What field point?
What wavelength or wavelength weighting?
What aperture?
What focus?
What direction: sagittal, tangential, radial, horizontal, vertical?
What frequency unit?
Is it diffraction MTF or geometric MTF?
Is it normalized to 1 at zero frequency?
What frequency range matters for the application?

The direction question is important.

Sagittal and tangential directions are defined relative to field geometry. They are not automatically the same as horizontal and vertical array slices unless the coordinate system is known.

The frequency unit is also important.

Cycles/mm, cycles/degree, and normalized frequency are not interchangeable.

A curve that looks strong on a normalized axis may not mean what you think at a sensor’s pixel pitch.

A better MTF reading habit is:

Do not ask only whether the curve is high.
Ask high at which frequency, for which field, in which direction, and under which wavelength and focus condition.

That is the difference between reading the curve and merely looking at it.

11. Always connect MTF back to PSF

MTF can feel abstract. When it does, walk backward.

MTF
← OTF
← PSF
← pupil function
← OPD
← ray tracing
← prescription

If MTF is low at high frequency, ask:

What does the PSF look like?

If sagittal and tangential MTF separate, ask:

Does the PSF or ray fan show directional blur?

If edge-field MTF is weak, ask:

What do the spot diagram, ray fan, and OPD map show at the edge field?

The MTF curve is powerful because it summarizes contrast transfer. It is limited because it compresses the spatial structure of blur.

Use it with PSF.

A healthy reading pair is:

PSF shows where point energy goes.
MTF shows how contrast survives by spatial frequency.

If you keep those two views together, MTF becomes much less mysterious.

12. Read optimization output as a before/after argument

When software says a design has been optimized, do not ask only:

Did the merit value go down?

Ask:

What variables changed?
What operands were included?
What targets were used?
What weights were used?
What constraints or bounds were active?
What improved?
What worsened?
What stayed unmeasured?

The merit value is a compressed score.

A lower merit value means the optimizer reduced the defined objective. It does not automatically mean the design is better in every meaningful way.

A careful optimization report should show:

prescription before and after
layout before and after
operand table before and after
spot diagrams
ray fans
wavefront maps
PSFs
MTFs
geometry constraints

This may seem like a lot. It is the minimum needed to understand what happened.

The optimizer follows the merit function, not your intention.

So after optimization, read the result as an argument:

The design changed in these ways.
These metrics improved.
These constraints remained acceptable.
These metrics did not improve or were not checked.
Therefore the design is better for this defined task.

That is much stronger than:

The optimized lens has lower merit.

13. Read differentiable optics outputs the same way

Differentiable optics adds gradients, but it does not remove the need for interpretation.

When reading a differentiable optimization result, ask:

What parameters were trainable?
What loss was optimized?
What physical model was used?
Were rays, wave optics, sensor sampling, and reconstruction included?
Were constraints smooth, hard, or absent?
Were gradients finite and stable?
Was the result validated outside the training loss?

A decreasing training loss is not a complete optical argument.

It is only one piece of evidence.

If a phase mask, freeform surface, or computational camera has been optimized end-to-end, still ask for classical checks:

PSF
MTF
field behavior
wavelength behavior
tolerance
manufacturability
sensor sampling
noise robustness
out-of-distribution images

Differentiable optics is exciting because it attaches gradients to the optical chain.

But the chain still needs to be physically meaningful.

A safe reading habit is:

Use gradients to search.
Use optical analysis to understand.
Use validation to trust.

That sentence is worth keeping.

14. The full reading checklist

Here is a practical checklist you can use when reading optical software outputs.

A. System definition

[ ] What is the lens prescription?
[ ] Are units clear?
[ ] Are surface signs correct?
[ ] Are materials correct?
[ ] Is the aperture stop defined?
[ ] Are semi-apertures or clear apertures reasonable?
[ ] Is the image plane where expected?
[ ] Are fields and wavelengths defined?

B. Layout and first-order checks

[ ] Does the layout look physically plausible?
[ ] Do rays pass through the expected apertures?
[ ] Are there ray failures?
[ ] Is effective focal length close to expectation?
[ ] Is f-number or NA correct?
[ ] Are pupil quantities plausible?
[ ] Is magnification, if relevant, correct?

C. Ray-based analysis

[ ] Which field and wavelength does the spot diagram show?
[ ] Is the scale clear?
[ ] Is the Airy disk shown or separately computed?
[ ] Does the ray fan explain the spot shape?
[ ] Are sagittal and tangential behaviors separated?
[ ] Is chromatic separation visible?

D. Wavefront analysis

[ ] What OPD reference is used?
[ ] Are piston, tilt, or defocus removed?
[ ] Are units in waves or length?
[ ] What wavelength is used?
[ ] Is the pupil mask clear?
[ ] Does the OPD pattern match the ray fan diagnosis?

E. PSF analysis

[ ] Was the PSF computed from a complex pupil function or another method?
[ ] Is it monochromatic or polychromatic?
[ ] Is the display linear or logarithmic?
[ ] Is normalization energy-based or peak-based?
[ ] Is the coordinate axis physical, angular, or pixel-based?
[ ] Is sampling adequate?
[ ] Does the Airy scale make sense?

F. MTF analysis

[ ] Is MTF computed from PSF / OTF or by another approximation?
[ ] Is it normalized at zero frequency?
[ ] What is the frequency unit?
[ ] Which field points are shown?
[ ] Which wavelengths or weights are used?
[ ] Which directions are shown?
[ ] Does the cutoff frequency make sense?
[ ] Are curves compared under identical settings?

G. Optimization analysis

[ ] What variables were released?
[ ] What operands were used?
[ ] What targets and weights were assigned?
[ ] What bounds and constraints were active?
[ ] Did the merit value decrease?
[ ] Which individual operands improved?
[ ] Did any important metric worsen?
[ ] Did geometry remain practical?
[ ] Was the result checked outside the optimized metrics?

This checklist is not meant to make you cautious to the point of paralysis.

It is meant to keep your confidence attached to evidence.

15. A shorter emergency checklist

When you have very little time, use this shorter version.

1. What system and configuration produced this output?
2. What exact quantity is plotted?
3. What are the units and normalization?
4. What assumptions or references are used?
5. What upstream calculation produced it?
6. What other plot should confirm or challenge it?
7. What would make this interpretation false?

The last question is especially useful.

A strong interpretation should be falsifiable.

For example:

Interpretation:
Edge MTF is low because off-axis astigmatism is spreading the PSF.

Checks:
Ray fan should show tangential/sagittal separation.
OPD should show field-dependent astigmatic structure.
PSF should be directionally spread.

If those checks fail, revise the interpretation.

This is how you move from plot reading to optical reasoning.

16. A small output-reading template

For your own work, you can write a short report for each major plot.

Use this template:

Output:
Configuration:
Input data:
Computation path:
Units and normalization:
Main observation:
Cross-check:
Limitation:
Next action:

For example:

Output:
Chapter 16 Cooke Triplet optimization report.

Configuration:
Three-field, three-wavelength Cooke Triplet analysis with the pinned optiland==0.6.0 companion code.

Input data:
Cooke Triplet prescription, selected radii and back focal distance as optimization variables.

Computation path:
Prescription -> ray tracing -> spot RMS merit -> bounded L-BFGS-B optimization.

Units and normalization:
Spot RMS merit in mm^2. The merit is lower when the selected-field RMS spot radii are lower.

Main observation:
The Chapter 16 companion run reduced spot RMS merit from 1.6625847615e-4 to 1.5389416611e-4, a 7.44% decrease.

Cross-check:
The optimization history is recorded in figures/chapter_16/cooke_optimization_history.csv, and the merit plot is recorded in figures/chapter_16/cooke_merit_optimization.png.

Limitation:
This run proves that the bounded optimization path executes and lowers the defined merit. It does not replace a full design acceptance review.

Next action:
Inspect layout, spots, ray fans, wavefront, PSF, MTF, and geometry constraints before accepting the design.

This kind of report may feel verbose at first. It is very effective.

It forces the output to sit inside the computation chain.

17. What you can now do

If you have followed the book, you now have several concrete abilities.

You can read a lens prescription

You know that a lens prescription is a data structure:

surfaces
radii
thicknesses
materials
apertures
aspheric terms
image plane

You know this is where computation begins.

You can trace one ray conceptually

You know a ray has an origin, direction, wavelength, and sometimes intensity or polarization state.

You know it propagates as:

$$\mathbf{r}(t)=\mathbf{r}_0+t\mathbf{d}$$

You know it intersects surfaces and refracts according to surface normals and refractive indices.

You can understand real versus paraxial rays

You know paraxial optics gives the first-order skeleton.

You know real rays reveal aberrations.

You know both matter.

You can interpret spot diagrams and ray fans

You know spot diagrams are ray intercept distributions.

You know ray fans show error structure across the pupil.

You know neither is the same as a diffraction PSF.

You can understand OPD and wavefront error

You know OPD connects geometric path calculation to wave phase.

You know OPD must be interpreted with wavelength, reference wavefront, and removed terms in mind.

You can build a pupil function

You know the pupil function is:

$$P(x,y)=A(x,y)\exp\left(i\frac{2\pi W(x,y)}{\lambda}\right)$$

You know it combines aperture amplitude and phase.

You can compute PSF from the pupil function

You know the PSF comes from the squared magnitude of a Fourier-transformed pupil field:

$$\mathrm{PSF}=|\mathcal{F}{P}|^2$$

You know sampling, padding, centering, and normalization matter.

You can compute MTF from PSF

You know:

$$\mathrm{OTF}=\mathcal{F}{\mathrm{PSF}}$$

$$\mathrm{MTF}=|\mathrm{OTF}|$$

You know the MTF curve is often a slice through a 2D frequency response.

You can understand optimization

You know a merit function is usually a weighted sum of operand errors.

You know variables, targets, weights, bounds, and constraints define what the optimizer can do.

You know lower merit is not automatically better design.

You can understand differentiable optics at the right level

You know differentiable optics attaches gradients to the optical computation chain.

You know autodiff can optimize through rays, waves, sensors, and reconstruction models if the computational path is differentiable.

You also know that gradients do not remove the need for physics, constraints, sampling, and validation.

That is a substantial toolkit.

18. What you should not conclude

It is just as important to know what this book has not claimed.

This book does not make you a professional optical designer by itself.

It does not replace years of design experience, manufacturing knowledge, tolerance analysis, stray light control, coating design, optomechanics, or system engineering.

It does not say minimal Python scripts are enough for production work.

It does not say Optiland or any open-source tool should blindly replace mature commercial tools in every engineering environment.

It does not say MTF is bad.

It does not say classical optical design is obsolete.

The claim is more specific and more useful:

If you can see the computation chain behind optical software outputs,
you can learn faster, debug better, and trust results for better reasons.

That is the promise.

19. How to continue learning

There are several natural next paths.

Path 1: Deeper geometrical optics

Study more complete aberration theory:

third-order aberrations
pupil aberration
field curvature
distortion
chromatic correction
stop shift effects

This will make ray fans and wavefront maps more meaningful.

Path 2: Fourier optics

Go deeper into:

coherent imaging
incoherent imaging
partial coherence
pupil autocorrelation
sampling theory
transfer functions
phase transfer

This will make PSF, OTF, and MTF feel less like isolated formulas.

Path 3: Optical design practice

Work through actual lens examples:

singlet
doublet
Cooke Triplet
Tessar-like forms
microscope objective simplifications
telephoto or wide-angle layouts

Analyze before optimizing. Then optimize. Then compare.

Path 4: Numerical methods

Study:

root finding
least squares
gradient descent
automatic differentiation
conditioning
local minima
parameter scaling

This will make optimization less mysterious.

Path 5: Computational imaging

Explore:

wavefront coding
diffractive optics
phase masks
sensor models
deconvolution
learned reconstruction
end-to-end differentiable design

This is where classical optics and modern computation meet.

Choose one path based on what you want to build.

The foundation from this book should help with all of them.

20. The final return to the MTF curve

Let us return to the original curve.

Imagine you are looking at an MTF plot now.

The old way to read it was:

The curve is high or low.
The lens is good or bad.

The new way is:

This curve is a selected slice through the magnitude of the OTF.
The OTF came from the Fourier transform of the PSF.
The PSF came from the complex pupil function.
The pupil function encoded aperture amplitude and OPD phase.
The OPD came from optical path differences through the system.
Those paths came from traced rays.
Those rays came from a prescription, materials, fields, wavelengths, aperture, and focus setting.

That is a much longer reading.

But it is also much more useful.

Now you can ask:

Which field is weak?
Which direction is weak?
Which frequency matters?
Is the weakness visible in the PSF?
Does the wavefront explain it?
Does the ray fan show the aberration structure?
Can optimization improve it?
What would be the cost of improving it?

The curve is no longer an endpoint.

It is a doorway back into the system.

That is the practical transformation this book was designed to create.

21. The quiet confidence of knowing the chain

There is a particular kind of confidence that comes from understanding a computation chain.

It is not the confidence of memorizing names.

It is not the confidence of pressing a button and hoping the software is right.

It is not the confidence of pretending complex things are simple.

It is quieter than that.

It sounds like this:

I may not know every detail yet,
but I know what inputs this result needs,
what calculation produced it,
what assumptions it depends on,
and what I should check next.

That is enough to keep going.

Optical design is full of detail. No single book removes that.

But once the chain is visible, detail becomes navigable.

You can place each new concept somewhere:

Is this about the prescription?
Is this about ray tracing?
Is this about paraxial structure?
Is this about aberration?
Is this about wavefront?
Is this about diffraction?
Is this about sampling?
Is this about optimization?
Is this about validation?

That map is the real outcome of the book.

22. The final checklist of capability

Here is the ending capability list promised by the blueprint.

After this book, you should be able to:

[ ] Read a basic lens prescription as a computational data table.
[ ] Explain how a ray is represented in code.
[ ] Describe how a ray intersects and refracts at a surface.
[ ] Understand why real rays differ from paraxial rays.
[ ] Interpret focal length, f-number, NA, and other first-order quantities.
[ ] Explain what a spot diagram computes.
[ ] Explain what a ray fan reveals.
[ ] Explain what OPD and wavefront error mean.
[ ] Convert OPD into pupil phase.
[ ] Build a complex pupil function from amplitude and phase.
[ ] Explain how a PSF is computed from the pupil function.
[ ] Explain how OTF and MTF are computed from the PSF.
[ ] Check sampling, normalization, FFT shift, and frequency-axis issues.
[ ] Understand what a merit function is.
[ ] Read optimization output as a before/after design argument.
[ ] Understand what automatic differentiation adds to optical optimization.
[ ] Ask better questions when optical software gives you a plot.

You do not need to be perfect at all of these immediately.

None of them should feel like magic anymore.

23. A final word on tools

This book used Optiland as an open reference point because open tools make the computation easier to inspect.

But the main character was never one tool.

The main character was the chain.

Commercial software, open-source libraries, and your own teaching scripts can all be valuable when you know what to ask of them.

A mature attitude toward tools is:

Use powerful software.
Respect its complexity.
Check its inputs.
Understand its outputs.
Reproduce small pieces when you need to learn.
Do not confuse the interface with the calculation.

That is the balance.

You do not need to rebuild every optical design tool from scratch.

But rebuilding small pieces teaches you what the tool is doing.

Once you have done that, the software output becomes less intimidating and more informative.

That is exactly where we wanted to arrive.

Chapter summary

This final chapter turned the book’s computation chain into a reading method for optical software outputs.

The central rule is:

Never read a plot alone.
Read the system, configuration, computation path, units, normalization, and assumptions behind it.

A lens output should be traced back through:

prescription
→ layout
→ paraxial analysis
→ real ray tracing
→ spot diagram and ray fan
→ OPD and wavefront
→ pupil function
→ PSF
→ OTF
→ MTF
→ merit function and optimization

Spot diagrams show ray intercepts. Ray fans show error structure across the pupil. OPD shows wavefront error relative to a reference. The pupil function encodes amplitude and phase. PSF shows point-source energy distribution. MTF shows contrast transfer by spatial frequency. Optimization changes variables to reduce a defined merit function. Differentiable optics attaches gradients to this same chain.

The final answer to the opening question is now clear:

An MTF curve is not from a software button.
It is the final visible result of a long optical computation chain.

Once you can see that chain, you can read optical software outputs with more patience, more skepticism, and more confidence.

Generated validation figure