Optics 036 · Imaging, instruments, and visual systems

Projector Optics Lab

Trace a finite source through a condenser, encoded LCD, finite projection pupil and tilted screen. Integrated screen textures expose focus and keystone, while exact circular angular overlap and panel quadrature expose field coupling and étendue saturation.

Upper / lower image width and central throw ratio

W_t,W_b,\mathrm{TR}_0

$0.50$

LCD distance and full-pupil blur envelope

s,b_{\mathrm{geom}}

$50\%$

Source / accepted étendue and coupling

G_s,G_a,\eta

$0.00\pi$

Model regime $\text{valid model regime}$

Ideal paraxial transmissive LCD and finite-source geometry, not a commercial DLP prescription. Diffraction, aberrations, fly-eye mixing, polarization, digital keystone and calibrated lumen output are omitted.

Physics tutorial

Audit the physics of Projector Optics Lab

BackgroundA projector illuminates a finite modulator and forms its real image on a screen. Moving the LCD changes the sharp image-plane distance through $\frac{1}{f}=\frac{1}{s}+\frac{1}{v}$ . A screen placed elsewhere receives defocused light. Its dimensions do not expand to fit the projected image.

Why it mattersWhy can a wider light source fail to brighten the screen, and why does a tilted screen lose focus as well as become trapezoidal?

Start with the essentials

Focus question: Why can a wider light source fail to brighten the screen, and why does a tilted screen lose focus as well as become trapezoidal?
One-sentence intuition: Brightness is limited by the source angular domain and the projection pupil, not by an arbitrary panel-size score. In the paraxial model $G_a\le\min(G_s,G_{\mathrm{capacity}})$ . Expanding a source of fixed radiance can eventually waste more flux without increasing accepted flux. Narrow illumination can also fail to reach off-axis panel points.

Core mathematical model

Finite conjugate and throw ratio

s=\frac{fv}{v-f},\quad W_0=\frac{wL}{s},\quad \mathrm{TR}_0=\frac{s}{w}

The LCD width is fixed. Only when the screen is perpendicular and the image is focused there does width equal the familiar far-field expression minus the finite-object correction. Focus and throw distance remain independent controls.

Source-to-angle mapping

\mathbf u_{\mathrm{after}}=-\frac{\mathbf y_s}{f_c},\quad q_s=\frac{r_s}{f_c},\quad G_s\simeq wh\pi q_s^2

The finite source lies at the condenser front focal plane. Each source position produces a parallel family after the condenser. Angular slope is a paraxial numerical-aperture proxy, not its exact sine definition. The condenser is large enough not to clip the permitted illumination domain.

Field-dependent angular coupling

\Omega_a(\mathbf y)=\frac{A\!\left(\mathcal D(\mathbf 0,a)\cap\mathcal D(\mathbf y,sq_s)\right)}{s^2},\quad G_a\simeq\int_{\mathrm{LCD}}\Omega_a\,\mathrm dA

Two circular domains at the projection-lens plane overlap. Their exact area controls local coupling; panel quadrature integrates total accepted étendue. The false-color panel shows accepted fraction of its incident angular domain, not screen irradiance.

Tilted-screen chief-ray map

y_{\mathrm{screen}}=-\frac{Ly_p}{s\cos\beta-y_p\sin\beta},\quad z_{\mathrm{screen}}=-\frac{Lz_p\cos\beta}{s\cos\beta-y_p\sin\beta}

These are actual ray-plane intersections, not a texture warp. The top and bottom have different scale. The sharp image plane remains perpendicular to the lens, so merely correcting image shape would not remove optical defocus.

Finite-pupil screen irradiance

E(\mathbf x)=\int_{\mathrm{pupil}}\mathcal L(\mathbf y_p)\frac{\cos\gamma_l\cos\gamma_s}{r^2}\,\mathrm dA_l

Trace each screen point backward through every pupil sample to the same finite LCD and source. Rejected rays contribute zero. One fixed radiance reference and gain are used for all controls; display white can clip. This is geometric irradiance in the stated ideal model, not a calibrated lumen measurement.

Common difficulties

More light-source area guarantees a brighter projection

Typical misconceptionKeep enlarging the emitter or digitally reshaping a trapezoid and the projector must become both bright and sharp.

Better mental modelFinite étendue capacity limits useful flux. A narrow source can underfill field bundles; a large source can overfill the lens. Digital shape correction does not move the optical focus plane. This model deliberately does not implement DLP micromirror or digital keystone algorithms.

Run the experiment

01
Scene 1: Focus and finite throw
Change throw distance without touching focus, then restore the focus distance to the screen center. Compare the LCD carriage shift, full-pupil blur envelope and actual screen texture.
What to observe: The optical cutaway is enlarged in millimetres; the upper room uses a uniform metre-scale mapping. The cyan rectangle marks a fixed central screen crop shown in the card. Cropping never changes ray coordinates, exposure or the real screen bounds.
02
Scene 2: Tilted screen and keystone
Tilt the screen through positive and negative angles. The trapezoid reverses, the gold sharp-plane intersection stays geometrically determined, and the screen keeps its real size.
What to observe: The LCD pattern is encoded upside down so its real inverted projection is upright. No independent screen-card rotation or keystone filter hides the optical inversion.
03
Scene 3: Illumination and étendue
Sweep source angular slope and pupil radius separately. Read both source and accepted étendue, and compare the coupling map with screen brightness; the map is not an exposure-normalized photograph.
What to observe: A tilted blur footprint need not have its centroid at the chief-ray hit. The width readout therefore uses independently traced chief rays, not a blur centroid.