Optics 036 · Imaging, instruments, and visual systems

Projector Optics Lab

Why can a bigger light source fail to brighten a projector, and why does a tilted screen also blur? The projection lens sets the limit. It accepts only a fixed étendue, roughly source area times the spread of ray angles. A larger source sends its extra light at angles the lens cannot take in, so that light is wasted. A tilted screen also cuts across the plane of sharp focus, so its edges blur as well as stretch into a trapezoid.

Trace a finite source through a condenser, an encoded LCD, a finite projection pupil and a tilted screen. Screen textures integrated from the rays show focus and keystone distortion. The exact overlap of circular angular regions and quadrature over the panel show 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 $\text{valid model regime}$

The model uses an ideal paraxial transmissive LCD and finite-source geometry; it is not the prescription of a commercial DLP projector. Diffraction, aberrations, fly-eye mixing, polarization, digital keystone correction and calibrated lumen output are not included.

Physics tutorial

Audit the physics of Projector Optics Lab

BackgroundA projector lights a small LCD panel, and its lens forms a real image of the panel on the screen. Moving the LCD changes where that image is sharp, following $\frac{1}{f}=\frac{1}{s}+\frac{1}{v}$ . A screen at any other distance receives a blurred image. The screen keeps its real size; it does not grow to fit the projected picture.

Why it mattersWhy can a bigger light source fail to brighten a projector, and why does a tilted screen also blur?

Start with the essentials

Focus question: Why can a bigger light source fail to brighten a projector, and why does a tilted screen also blur?
One-sentence intuition: Screen brightness is capped by two things: the range of angles the source fills and the opening of the projection lens. In the paraxial model $G_a\le\min(G_s,G_{\mathrm{capacity}})$ . Past some point, a larger source of the same radiance only spills more light outside the lens, and the accepted flux stops growing. Illumination that is too narrow can miss the off-axis parts of the panel.

Core mathematical model

LCD position, picture width and throw ratio

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

The LCD has a fixed width. Only when the screen is square to the lens and the image is in focus on it does the picture width equal the familiar far-field value minus a correction for the finite object distance. Focus and throw distance are still set separately.

How source position becomes illumination angle

\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 sits in the front focal plane of the condenser, so each point of the source leaves the condenser as a bundle of parallel rays. The angular slope stands in for numerical aperture in the paraxial limit; it is not the exact sine definition. The condenser is large enough that it never clips the allowed range of illumination angles.

How much light each panel point sends into the lens

\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

At the projection-lens plane, the light cone from one panel point and the lens opening are two circles. The exact area where they overlap sets how much of that point's light the lens accepts, and integrating over the panel gives the total accepted étendue. The false-color panel map shows what fraction of each point's incoming angles is accepted; it is not the brightness on the screen.

Where chief rays land on a tilted screen

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 formulas give where each chief ray actually meets the tilted screen plane. The top and bottom of the picture end up at different scales. The plane of sharp focus stays perpendicular to the lens axis, so fixing the picture's shape alone would not remove the optical blur.

Screen irradiance from a finite lens opening

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

For each screen point, rays are traced backward through every sample of the lens opening to the same finite LCD and source. A ray that is blocked contributes nothing. All settings share one fixed radiance reference and one gain, so the brightest areas can clip to white. The result is geometric irradiance in this ideal model, not a calibrated measurement in lumens.

Common difficulties

A bigger light source always gives a brighter picture

Typical misconceptionMake the emitter bigger and the picture gets brighter; straighten the trapezoid digitally and it is sharp again.

Better mental modelThe lens accepts only a limited étendue, and that caps the useful flux. A narrow source can underfill the ray bundles of the field points; a large source can overfill the lens. Correcting the shape digitally does not move the plane of optical focus. This model leaves out DLP micromirrors and digital keystone algorithms on purpose.

Run the experiment

01
Scene 1: Focus and finite throw
Change the throw distance and leave focus alone. Then bring the focus distance back to the screen center. Compare how far the LCD carriage moves, the blur envelope from the full lens opening and the picture on the screen.
What to observe: The optical cutaway is drawn enlarged, in millimetres, and the room above uses one uniform scale in metres. The cyan rectangle marks the fixed central part of the screen shown in the image card. Cropping does not change ray coordinates, exposure or the real edges of the screen.
02
Scene 2: Tilted screen and keystone
Tilt the screen through positive and negative angles. The trapezoid flips direction. The gold line, where the plane of sharp focus cuts the screen, follows from the geometry, and the screen keeps its real size.
What to observe: The LCD pattern is stored upside down, so the lens's inverted image comes out upright on the screen. The screen card shows that image as it is, with no extra rotation and no keystone filter.
03
Scene 3: Illumination and étendue
Sweep the source angular slope and the pupil radius one at a time. Read the source étendue and the accepted étendue, and compare the coupling map with the screen brightness. The coupling map is not an auto-exposed photograph.
What to observe: On a tilted screen, the center of a blur spot need not sit where the chief ray lands. The width readout therefore uses separately traced chief rays instead of blur centers.