Optics 038 · Imaging, instruments, and visual systems

VR Headset Optics

Why can a VR headset change depth while focus stays fixed, and why does a small eye shift cut off the corners? Stereo depth and focus come from different places. Depth comes from the difference between the two eyes' pictures, which can change freely. Focus is set by the lens, which puts a virtual image of the display at one fixed distance. The lens delivers the whole picture only inside a small zone, the eye-box; move the pupil out of it and rays from the corners no longer enter the eye.

A pair of eyepieces, stereo displays and movable finite pupils determine analytically which parts of the field each eye can see. Two computed angular views separate stereo disparity from optical focus, and a pupil-plane map shows the full-field eye box and where the view goes black.

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 $\text{valid model regime}$

Model checked against analytic results

VR Headset Optics

Display-to-eyepiece distance (moves the actual screen)

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 change the stereo depth without moving the optical focus. Then move the display to bring the stereo depth and the optical focus together. Shift the eye's finite pupil and watch the corners of the view disappear before the center. The green map is a slice of the eye box for the full field, not just the size of the lens opening.

The eyepieces are ideal, paraxial, 45 mm thin lenses, not a commercial wide-angle VR design. The angular maps are geometric calculations, not retinal photographs. Fresnel or pancake folded optics, distortion, pupil swim, chromatic artifacts and human comfort are not simulated.

Physics tutorial

Audit the physics of VR Headset Optics

BackgroundThe two displays show the marker in slightly different directions, but both optical channels focus at the same depth. When a display sits inside the focal length of a positive eyepiece, the rays leave diverging, and the focus the eye needs is $A=\frac{1000c}{1+ec},\qquad c=\frac1s-\frac1f$ . In the curvature relation, distances are in millimetres, and the focus demand comes out in dioptres. The eye relief has to be included. With the display at the focal plane, the outgoing bundles are parallel and the focus demand is exactly zero.

Why it mattersWhy can a VR headset change depth while focus stays fixed, and why does a small eye shift cut off the corners?

Start with the essentials

Focus question: Why can a VR headset change depth while focus stays fixed, and why does a small eye shift cut off the corners?
One-sentence intuition: Moving the stereo marker in depth does not move the optical focus. The mismatch between the two cues is $\Delta_{AV}=\left|A-\frac1{Z_V}\right|$ . This is a geometric mismatch between depth cues; it does not predict anyone’s fatigue and is not a comfort score. To change the optical focus you have to move the screen itself, because stereo disparity does not refocus anything.

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 virtual image lies on the display side of the eyepiece. Focal depth is measured from the pupil, not from the lens. At zero curvature, take the smooth limit where the demand goes to zero and the focus is at infinity. The focal length is fixed at 45 mm, and 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}

A ray from a screen point travels to where it actually meets the lens, takes the ideal thin-lens change in slope, and continues to the pupil plane. The viewing direction has the opposite slope to the outgoing ray. The fixed angular cards use these directions through the pupil center, shaded by the analytic fraction that enters the pupil; they do not simulate blur on the retina.

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 lights a circular footprint at the eye plane. The analytic overlap of that circle with the finite eye pupil gives the fraction of the pupil that receives light; it is not source power or calibrated luminance. A corner of the field can disappear while the center still gets through. The pink example rays stop where they miss the lens opening.

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 a square field is a corner. A position counts as green only if the bundle from every corner covers the whole finite pupil, not just its center. The four cyan outlines are the corner-bundle footprints. The white circle is the chosen pupil at its real offset and diameter. In the amber zone the central field gets through, but corners may be lost.

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 applies only along the horizontal center line of the full-field eye-box slice. If the existence condition fails, the width is reported as zero. Changing the eye relief changes how far apart the footprints are, and it often tightens the tolerance. The map shows the whole two-dimensional slice, 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 two channels are 64 mm apart. The marker is ideally recalibrated to the current pupil-center positions so that it shows the chosen target depth. It changes the directions seen by the two eyes but leaves the wavefront curvature unchanged. This is a well-defined geometric stereo stimulus; the Lab does not implement 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 sets the cue for where the eyes converge; the spacing between display and lens sets the cue for where they focus. The two stay mismatched even when the marker looks three-dimensional. The model sets no universal discomfort threshold and does not predict anyone’s visual health.

Run the experiment

01
Scene 1: Collimated optical focus
Start with collimated focus. Move the stereo-marker depth toward 0.4 m. The pink targets move apart and the stereo-demand bar rises, while the optical-demand bar stays at zero.
What to observe: All optical parts and ray segments are drawn on one uniform millimetre scale. Each card shows a fixed window in angle or in the pupil plane, and the cards stay in view when you orbit the scene on a desktop or touch screen.
02
Scene 2: Stereo depth versus focus
Use the depth-versus-focus preset. Move the screen inward until the two demand bars meet. Watch the screen position and the focal-depth readout change; the focus moves because the screen moves, not because the image is rescaled.
What to observe: Full-field tolerance is stricter than transmission on axis. The horizontal admitted field covers every point on the horizontal center line of the screen whose bundle overlaps the pupil at all; it does not promise high brightness across the whole image.
03
Scene 3: Finite-pupil eye-box
Choose eye-box. Move both pupils through the green region, across the amber edge and into the black area outside. Then vary the lens radius, pupil diameter and eye relief one at a time.
What to observe: This first-order model leaves out the Fresnel grooves of commercial lenses, pancake polarization folds, off-axis prescriptions, distortion, colour fringes, screen-door texture and pupil swim. Combinations with large slopes are flagged. The drawn eye shell only shows the pupil opening; it does not model retinal anatomy.