Optics 029 · Imaging, instruments, and visual systems

Human Eye & Corrective Optics

What lens corrects short sight, long sight, presbyopia or astigmatism? The lens that puts the focus back on the retina. Short sight needs a negative lens, long sight and presbyopia a positive one, and astigmatism a cylindrical one. Glasses sit a little in front of the eye, so their power is adjusted for that distance. The correction works when a point of light again forms a tight spot on the retina.

A transparent three-dimensional reduced eye combines spectacle vertex conversion, crystalline-lens geometry that changes with accommodation, astigmatic ray tracing in two meridians and live scalar diffraction at the pupil. Every scene shows the uncorrected and corrected retinal PSFs side by side, next to the absorbing retina.

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 remaining focus error

S_0, |\Delta z|

$0.00\pi$

Model regime $\text{valid model regime}$

The model is a centered 60 D reduced eye in paraxial optics, with a fixed 12 mm spectacle vertex distance and a monochromatic 550 nm Fraunhofer PSF for a circular pupil. Higher-order aberrations of the eye, chromatic effects, retinal and neural processing, pathology and individual biometry are not included. This is not a prescription or a diagnosis.

Physics tutorial

Audit the physics of Human Eye & Corrective Optics

BackgroundYou cannot simply add a spectacle prescription to the eye's own power. The lens sits in front of the cornea and changes each ray's height before the eye's power acts, so the equivalent power at the cornea is $F_{\mathrm{cornea}}=\frac{F_{\mathrm{spec}}}{1-d_vF_{\mathrm{spec}}}$ . The Lab carries each ray across the vertex distance shown and checks separately where it meets the retina.

Why it mattersWhat lens corrects short sight, long sight, presbyopia or astigmatism?

Start with the essentials

Focus question: What lens corrects short sight, long sight, presbyopia or astigmatism?
One-sentence intuition: Rays crossing at one point do not by themselves prove a sharp image. For the same circular pupil and the eye's quadratic phase error, this Lab computes 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 show this computed diffraction intensity before and after correction, and their on-axis Strehl readouts measure how much the correction recovers.

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 invert each other exactly to first order. The difference matters more for stronger lenses, so the model reports spectacle-plane and corneal-plane powers separately.

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 each ray at its fixed plane. If the Gaussian focus lies behind the retina, the scene only marks its computed position; no ray is drawn passing through the retina.

Accommodation needed and the Navarro lens geometry

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

In the near scene, the eye accommodates only up to the selected amplitude. The crystalline lens visibly steepens, following the published logarithmic relation for its front radius. Whatever vergence is still missing comes from a plus addition, traced exactly.

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 has residual powers of equal size and opposite sign on either side of its spherical equivalent. Its two real line foci lie one in front of the retina and one behind it. The toric lens applies the opposite power in each meridian, 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 pupil diameter and wavelength shown. A larger pupil shrinks the diffraction-limited Airy pattern but enlarges the geometric blur from a given defocus, so pupil size is a genuine 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 that leaves out several things on purpose. A real eye exam needs the patient's subjective responses, binocular balance, individual biometry, ocular health, lens positioning and clinical judgment. None of these is in 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: A minus distance prescription means the relaxed eye is too strong for the distance to its retina, so the uncorrected focus lies in front of the retina. A plus prescription means the opposite: the computed focus lies behind the retina.
02
Scene 2: Presbyopic accommodation
Switch to the reading target at forty centimetres. Lower the accommodation amplitude below what the target needs, then watch the crystalline lens relax, the uncorrected focus move behind the retina and the exactly solved plus addition bring the cyan rays back onto the retina.
What to observe: With enough accommodation amplitude, the exact reading addition drops to zero. Below that threshold the addition rises smoothly. The Navarro front radius shrinks and the crystalline lens curves more, but only as far as the available accommodation allows.
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: The two astigmatic foci are symmetric in power, not in distance, so their positions along the axis are not symmetric. Only a correction matched to both meridians brings both residuals to zero at once.