Optics 029 · Imaging, instruments, and visual systems

Human Eye & Corrective Optics

A transparent three-dimensional reduced eye couples spectacle vertex conversion, accommodation-dependent crystalline-lens geometry, two-meridian astigmatic ray tracing, and live scalar pupil diffraction. Every scene compares uncorrected and corrected retinal PSFs beside the absorbing retina instead of using decorative blur.

Prescription, accommodation, or meridional error

F_{\mathrm{spec}}, A, \Delta F_{x,y}

$0.50$

Focus position or lens geometry

z_f, z_R, R_3

$50\%$

Retinal PSF recovery and focus closure

S_0, |\Delta z|

$0.00\pi$

Model regime $\text{valid model regime}$

Centered 60 D reduced-eye paraxial optics with a fixed 12 mm spectacle vertex and a monochromatic 550 nm circular-pupil Fraunhofer PSF. Higher-order ocular aberrations, chromatic effects, retinal and neural processing, pathology, and individual biometry are omitted; this is not a prescription or diagnosis.

Physics tutorial

Audit the physics of Human Eye & Corrective Optics

BackgroundAn eyeglass prescription is not simply added at the eye plane. A lens held in front of the cornea changes ray height before the eye power acts, so the equivalent corneal-plane power is $F_{\mathrm{cornea}}=\frac{F_{\mathrm{spec}}}{1-d_vF_{\mathrm{spec}}}$ . The apparatus propagates each ray across the displayed vertex distance and independently verifies where it intersects the retinal plane.

Why it mattersWhich correction returns the retinal point-spread function for each refractive error?

Start with the essentials

Focus question: Which correction returns the retinal point-spread function for each refractive error?
One-sentence intuition: A sharp ray crossing alone is not an image-quality proof. For the same circular pupil and quadratic ocular phase, this Lab evaluates the scalar retinal point-spread function $\mathrm{PSF}(x,y)=\left|\mathcal F\!\left\{P(\xi,\eta)e^{\mathrm i\phi(\xi,\eta)}\right\}\right|^2$ . The warm and cyan cards therefore show computed diffraction intensity before and after correction, while their on-axis Strehl readouts audit recovery numerically.

Core mathematical model

Spectacle vertex conversion

F_c=\frac{F_s}{1-d_vF_s},\qquad F_s=\frac{F_c}{1+d_vF_c}

The two expressions are exact first-order inverses. This matters increasingly for stronger powers; the model keeps spectacle-plane and corneal-plane values separate instead of labeling them as the same lens.

Reduced-eye ray transfer

\binom{y_R}{u_R}=\mathbf P(z_R)\mathbf L(F_{\mathrm{eye}})\mathbf P(d_v)\mathbf L(F_s)\binom{y_0}{u_0}

The retina absorbs the ray at its fixed plane. A focus beyond it is shown only as a labeled Gaussian construction location; no physical ray is drawn propagating through the retinal detector.

Accommodation demand and Navarro geometry

A_{\mathrm{used}}=\min(A_{\mathrm{amp}},s^{-1}),\quad R_3(A)=10.2-1.75\ln(A+1)\ \mathrm{mm}

The near scene caps active accommodation by the selected amplitude. Its crystalline lens becomes visibly steeper using the published logarithmic anterior-radius relation; any remaining vergence demand is closed by an exactly traced plus addition.

Two-meridian astigmatism

\Delta F_x=+\frac{C}{2},\qquad \Delta F_y=-\frac{C}{2},\qquad z_{x,y}=\frac{1}{F_0+\Delta F_{x,y}}

The teaching eye places equal and opposite residual powers around its spherical equivalent. The two real line foci straddle the retina; the displayed toric lens applies the opposite meridional powers after vertex conversion.

Pupil diffraction and defocus phase

\phi(\xi,\eta)=-\frac{\pi}{\lambda}\left(\Delta F_x\xi^2+\Delta F_y\eta^2\right),\qquad d_A=2.44\frac{\lambda z_R}{D_p}

The numerical pupil transform uses the displayed pupil diameter and wavelength. Enlarging the pupil shrinks the diffraction-limited Airy scale but enlarges a given geometric defocus footprint, so pupil size creates a real tradeoff.

Common difficulties

Reading the rendering as a medical prescription

Typical misconceptionIf the PSF card looks sharp, the displayed power is enough to prescribe lenses for a person.

Better mental modelThis is a centered reduced-eye experiment with selected omissions. Real refraction requires subjective responses, binocular balance, individual biometry, ocular health, lens positioning, and clinical judgment that are intentionally outside this Lab.

Run the experiment

01
Scene 1: Myopia and hyperopia
Start with distance refraction. Sweep the signed spectacle prescription through minus, zero, and plus powers. Compare the lens surface, uncorrected focus marker, corneal-equivalent readout, retinal ray height, and warm PSF before judging myopia or hyperopia.
What to observe: Minus distance prescriptions correspond to an eye whose relaxed power is too strong for its retinal distance, so the uncorrected focus lies in front of the retina. Plus prescriptions encode the opposite case and move that construction behind the absorbing surface.
02
Scene 2: Presbyopic accommodation
Switch to the forty-centimetre reading target. Lower accommodation amplitude below the target demand, then watch the crystalline lens relax, the uncorrected focus move behind the retina, and the exactly solved plus addition return cyan rays to the detector.
What to observe: At sufficient accommodation amplitude the exact reading addition falls to zero. Below that threshold the addition rises continuously, while the Navarro anterior radius decreases and the displayed crystalline lens becomes more curved only up to the available accommodation.
03
Scene 3: Astigmatic correction
Open astigmatic correction. Increase cylinder and orbit the eye to separate the warm horizontal and vertical ray fans. Then enlarge the pupil and compare the two-axis uncorrected PSF with the compact toric-corrected result.
What to observe: Astigmatic principal foci separate symmetrically in power rather than in distance. Their axial distances are therefore not symmetric, and only the matched two-meridian correction makes both corrected residuals close simultaneously.