Optics 027 · Imaging, instruments, and visual systems

Lens Aberration Clinic

How do you identify a lens aberration from its blur and fix it without creating another? Start from the shape of the blur. Spherical aberration leaves a round halo, coma a comet-shaped tail, and astigmatism a line-like smear. Each shape comes from its own term in the wavefront error, and the matching correction cancels that term. Judge the fix across the whole lens opening, not by one spot size, so the error is not simply moved into another aberration.

A 2D optical diagnostics bench turns Seidel wavefront terms into analytic transverse ray fans, spot diagrams at physical scale, a signed exit-pupil map and a treatment sweep for spherical aberration, coma, astigmatism, field curvature and primary chromatic correction.

Wavefront RMS and Maréchal estimate

\sigma_W,\ \widehat S_M

$0.50$

Geometric RMS and full spot radii

r_{\mathrm{RMS}},\ r_{100}

$50\%$

Treatment gain and Airy reference

\Delta r_{\mathrm{RMS}},\ r_A

$0.00\pi$

Model regime $\text{valid model regime}$

Physics tutorial

Audit the physics of Lens Aberration Clinic

BackgroundFirst-order optics predicts a single Gaussian image point, but a real wavefront can depart from the ideal reference sphere. This Clinic describes that departure with the normalized third-order expansion $W=W_{040}\rho^4+W_{131}H\rho^3\cos\phi+W_{222}H^2\rho^2\cos^2\phi+W_{220}H^2\rho^2$ . Each term has its own symmetry across the pupil and leaves a recognizable pattern in the ray fans and the spot diagram, so you can identify it from that pattern instead of guessing from a round blur.

Why it mattersHow do you identify a lens aberration from its blur and fix it without creating another?

Start with the essentials

Focus question: How do you identify a lens aberration from its blur and fix it without creating another?
One-sentence intuition: Where a ray lands in the spot diagram is set by the local slope of that same wavefront, scaled by the reference focal length and pupil radius: $\Delta\mathbf r=f\frac{\lambda}{R_p}\nabla_{\!p}W$ . The pupil map, the two ray fans, the spot cloud, the RMS radius and the treatment sweep all come from one wavefront, so any disagreement between them would point to a computing error.

Core mathematical model

Third-order wavefront

W(\rho,\phi)=W_{040}\rho^4+W_{131}H\rho^3\cos\phi+W_{222}H^2\rho^2\cos^2\phi+W_{220}H^2\rho^2

Spherical aberration is rotationally symmetric. Coma is odd along the field direction. Astigmatism focuses the tangential and sagittal fans at different places. Field curvature acts on a flat sensor as defocus that changes with field position.

Wavefront slope to ray error

\Delta\mathbf r=f\frac{\lambda}{R_p}\left(\frac{\partial W}{\partial x_p},\frac{\partial W}{\partial y_p}\right)

The Lab differentiates the polynomial analytically at every sampled pupil point. Before any point is drawn, a finite-difference check confirms that gradient independently.

Small-error Strehl estimate

\widehat S_M=\exp\!\left[-\left(2\pi\sigma_W\right)^2\right]

Piston and tilt are removed before the RMS wavefront error over the pupil is calculated. The result is a Maréchal estimate, and the Lab marks where that approximation stops holding. It is not a computed diffraction PSF.

Primary achromat condition

\Phi_1/V_1+\Phi_2/V_2=0

Two elements of opposite power, made from glasses with different Abbe numbers, can bring the F-line and C-line foci back together. The d line can still be left with secondary spectrum, which stays visible in the last treatment scene.

Common difficulties

Calling every small spot diffraction limited

Typical misconceptionIf the geometric spot fits inside the Airy reference circle, the displayed point cloud is already a diffraction point-spread function.

Better mental modelA spot diagram shows where rays hit; an Airy pattern shows coherent diffraction intensity. This Lab draws only the radius of the first dark ring, as a scale reference, and reports the Maréchal estimate separately.

Run the experiment

01
Scene 1: Spherical and coma diagnosis
Open spherical and coma diagnosis. Increase the aberration magnitude and compare the radial spherical fan with the lopsided coma tail. Then raise treatment and watch the same points draw in while the plot scale stays fixed.
What to observe: Pure spherical aberration stays centered and rotationally symmetric. Coma sends different parts of the pupil to different places and leaves a comet-shaped cloud, even after the centroid is removed.
02
Scene 2: Astigmatism and field curvature
Switch to astigmatism and field curvature. Compare the solid tangential fan with the dashed sagittal fan and verify that the field-flattener prescription reduces their separation at the fixed image plane.
What to observe: Astigmatism can make one ray fan look corrected while the perpendicular fan stays broad. A single RMS number cannot reveal that direction-dependent error on its own.
03
Scene 3: Achromat and asphere treatment
Select achromat and asphere treatment. Sweep treatment from singlet toward doublet and verify that the blue F and red C spot families reunite while a smaller green d-line residual remains visible.
What to observe: Primary achromatization can match two selected wavelengths without eliminating every wavelength-dependent focus shift. More elements or anomalous dispersion are required to control the residual spectrum further.