Optics 038 · Imaging, instruments, and visual systems

VR Headset Optics

A paired eyepiece workbench links real stereo displays and moving finite pupils to analytic field admission. Two computed angular views separate stereo disparity from optical focus; a pupil-plane map exposes the full-field eye-box and blackout.

Optical / stereo demand and mismatch

A,V,\Delta_{AV}

$0.50$

Horizontal field and axial pupil admission

\theta_{\mathrm{panel}},\theta_{\mathrm{admitted}},T_0

$50\%$

Focus depth / full-field eye-box / corner admission

L_A,b_{\mathrm{full}},T_{\mathrm{corner,min}}

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

VR Headset Optics

Display-to-eyepiece distance (actual screen movement)

s

45\,\mathrm{mm}

Eye relief (eyepiece to pupil plane)

e

18\,\mathrm{mm}

Horizontal translation of both eye pupils

p_x

0\,\mathrm{mm}

Actual eyepiece clear radius

a

16\,\mathrm{mm}

Actual eye-pupil diameter

2r

4\,\mathrm{mm}

Stereo-marker depth (optical focus unchanged)

Z_V

5\,\mathrm m

First move stereo depth without moving optical focus. Then move the real display to bring the two demands together. Shift the finite eye pupil and watch corners disappear before the center. The green map is a full-field eye-box slice, not just the lens opening.

Ideal paraxial 45 mm thin eyepieces, not a commercial wide-angle VR prescription. Angular maps are geometric, not retinal photographs. Fresnel/pancake folds, distortion, pupil swim, chromatic artifacts and human comfort are not simulated.

Physics tutorial

Audit the physics of VR Headset Optics

BackgroundTwo displays encode different marker directions, but both optical channels share one focal depth. A display inside a positive eyepiece focus produces divergent rays with focal demand $A=\frac{1000c}{1+ec},\qquad c=\frac1s-\frac1f$ . Here distances in the curvature relation use millimetres; demand uses dioptres. Eye relief must be included. At the focal plane, outgoing bundles are parallel and focal demand is exactly zero.

Why it mattersWhy can stereo depth move while optical focus stays fixed, and why does a small eye shift erase the display corners?

Start with the essentials

Focus question: Why can stereo depth move while optical focus stays fixed, and why does a small eye shift erase the display corners?
One-sentence intuition: Moving a stereo marker does not move optical focus. Their demand mismatch is $\Delta_{AV}=\left|A-\frac1{Z_V}\right|$ . It is a geometric cue mismatch, not a prediction of a person’s fatigue or a comfort score. Move the actual screen to adjust optical focus instead of pretending stereo disparity performs refocusing.

Core mathematical model

Finite virtual focus and eye relief

s'=-\frac1c,\quad K=1+ec,\quad L_A=e+\frac1c,\quad A=\frac{1000c}{K}

The signed image lies on the display side of the eyepiece; focal depth is measured from the pupil, not the lens. At zero curvature use the continuous zero-demand limit, with focus at infinity. Fixed focal length is 45 mm; both displays are 24 mm squares.

Same rays, same angular map

u_{\mathrm{out}}=\frac{h_\ell-h_s}{s}-\frac{h_\ell}{f},\quad h_p=Kh_\ell-\frac es h_s,\quad q=\frac{h_s/s-cp_x}{K}

Propagate a screen point to the actual lens intersection, apply the ideal thin-lens slope kick, then reach the pupil plane. The viewing slope is the negative of outgoing propagation slope. The fixed angular cards use these pupil-center directions and analytic pupil-admission shading, not a retinal blur simulation.

Finite pupil and field-dependent admission

R=Ka,\quad \mathbf b=-\frac es\mathbf h_s,\quad T(\mathbf h_s)=\frac{\operatorname{area}\!\left(B_R(\mathbf b)\cap B_r(\mathbf p)\right)}{\pi r^2}

Each display point creates a circular bundle footprint at the eye plane. Its analytic circle intersection measures the admitted fraction of the finite eye pupil, not source power or calibrated luminance. A field corner can vanish while the center still passes. Pink illustrative rays end where they miss the lens opening, not at a fictitious transparent mirror.

Whole-field eye-box at one axial slice

\max_{\mathbf h_s\in\{(\pm H,\pm H)\}}\|\mathbf p+\tfrac es\mathbf h_s\|+r\le Ka

The farthest point of the square field is a corner. Green requires every corner bundle to cover the entire finite eye pupil, not just its center. The four cyan outlines are corner-bundle footprints. The white circle is the selected pupil at the real offset and diameter; amber admits the central field but may lose corners.

Horizontal full-field tolerance

b_{\mathrm{full}}=\sqrt{(Ka-r)^2-\beta^2}-\beta,\quad \beta=\frac{eH}{s},\quad Ka-r\ge\sqrt2\,\beta

This half-width is only the horizontal center line of the full-field eye-box slice. If the existence condition fails, the reported width is zero. Changing eye relief changes the footprint separation and often tightens tolerance. The map shows the complete two-dimensional slice; it is not a three-dimensional eye-box volume.

Stereo cue is not optical refocusing

q_\pm=-\frac{x_\pm}{1000Z_V},\quad h_{s,\pm}=s(Kq_\pm+cp_x),\quad V=\frac1{Z_V}

The channels are 64 mm apart. The marker is ideally recalibrated to current pupil-center positions to encode the selected axial target. It changes binocular directions without changing wavefront curvature. This is a defined geometric stereo stimulus, not implemented eye tracking, neural fusion or a gaze-driven varifocal controller.

Common difficulties

A nearby stereo object automatically becomes optically near

Typical misconceptionIf the two eye images indicate a nearby target, the display must also focus at that distance.

Better mental modelDisparity controls vergence stimulus; display-to-lens spacing controls focal stimulus. Their mismatch persists even when the marker looks three-dimensional. The model does not assign a universal discomfort threshold or predict individual visual health.

Run the experiment

01
Scene 1: Collimated optical focus
Start with collimated focus. Move stereo-marker depth toward 0.4 m: the pink targets separate and the stereo-demand bar rises, while the optical-demand bar remains zero.
What to observe: All optical hardware and ray segments share a uniform millimetre mapping. Cards have explicitly fixed angle or pupil-plane windows and remain in view after orbiting on desktop or touch screens.
02
Scene 2: Stereo depth versus focus
Use the depth-versus-focus preset. Move the actual screen inward until the two demand bars meet. Observe the changed screen position and focal-depth readout; no focus plane is moved by a fake image scaling.
What to observe: Full-field tolerance is stricter than axial transmission. Horizontal admitted field reports the screen center meridian with any nonzero pupil overlap, not guaranteed high brightness across a whole image.
03
Scene 3: Finite-pupil eye-box
Choose eye-box. Translate both pupils through the green region, amber edge and black outside. Then vary the actual lens radius, pupil diameter and eye relief independently.
What to observe: This first-order model has no commercial Fresnel grooves, pancake polarization folds, off-axis prescription, distortion, chromatic fringe, screen-door texture or pupil swim. Large-slope combinations are flagged; the drawn eye shell illustrates a pupil gate, not retinal anatomy.