Optics 027 · Imaging, instruments, and visual systems

Lens Aberration Clinic

A light 2D optical-diagnostics bench turns Seidel wavefront terms into analytic transverse ray fans, physically scaled spot diagrams, 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 one Gaussian image point, but a real wavefront can depart from its reference sphere. This Clinic uses 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 a distinct pupil symmetry, so its ray-fan and spot-diagram signature can be diagnosed rather than guessed from a blur circle.

Why it mattersWhich optical correction cures each spot-diagram signature without creating a worse one elsewhere?

Start with the essentials

Focus question: Which optical correction cures each spot-diagram signature without creating a worse one elsewhere?
One-sentence intuition: A geometric spot is the local slope of that same wavefront, converted through the reference focal length and pupil radius: $\Delta\mathbf r=f\frac{\lambda}{R_p}\nabla_{\!p}W$ . The pupil map, two ray fans, spot cloud, RMS radius, and treatment sweep therefore cannot disagree without exposing an implementation 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 across field orientation, astigmatism separates tangential and sagittal behavior, and field curvature acts as field-dependent defocus at a flat sensor.

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 implementation differentiates the polynomial analytically at every sampled pupil coordinate. The finite-difference benchmark independently checks that gradient before any point is drawn.

Small-error Strehl estimate

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

Piston and tilt are removed before the pupil RMS is calculated. The interface calls this a Maréchal estimate and marks its approximation boundary instead of presenting it as a computed diffraction PSF.

Primary achromat condition

\Phi_1/V_1+\Phi_2/V_2=0

Opposite powers in glasses with different Abbe numbers can reunite the F and C line foci. The d line may retain secondary spectrum, which remains visible in the final 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 contains ray intercepts; an Airy pattern contains coherent diffraction intensity. This Lab overlays only the first-zero radius 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 aberration magnitude, compare the radial spherical fan with the asymmetric coma tail, then raise treatment to watch the same points contract without changing the plot scale.
What to observe: Pure spherical structure stays centered and rotationally symmetric, whereas coma shifts pupil sectors into a comet-like density 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 orthogonal fan remains broad. One scalar RMS number cannot identify that directional failure by itself.
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.