Sharp image on a fixed screen: defocus, not a bad lens
A student is asked to project a translucent grid (grid spacing 1.0 mm) onto a wall using a plano-convex lens of focal length 10.0 cm and clear diameter 4.0 cm, at a linear magnification of 10. The wall bracket is fixed at 100 cm from the lens, and she places the grid 11.0 cm from the lens on the axis, setting the distance with a ruler to plus or minus 1 mm. The projected image is recognisable but blurred: the blur is uniform across the field, about 3 mm wide, and does not depend on where on the grid she looks. Masking the lens down to a 2.0 cm aperture shrinks the blur to about half, in proportion to the aperture, which she takes as proof that the lens is badly figured (spherical aberration), and she asks for a better-corrected lens. A technician suggests checking the geometry before blaming the optics.
1. Use the lens equation to find where the image actually forms, and estimate the blur expected from the screen position alone. Does this match the observation?
Read the worked answer
Thin lens, real object:
1/v = 1/f - 1/u = 1/10.0 - 1/11.0 = (11 - 10)/110 = 1/110.
v = 110 cm, m = v/u = 10 (the intended magnification is achieved).
The screen is at 100 cm, i.e. 10 cm short of the image plane. The cone of rays from an axial object point has converged to a diameter
b = D |delta_v|/v = 40 mm x 100/1100 = 3.6 mm.
This matches the observed 3 mm blur, and it explains the two qualitative features: the blur is the same everywhere in the field (defocus scales with distance from the image plane, not with field angle) and it is a simple magnified blur of the grid rather than a comet-like flare. No optical defect is needed to explain the observation; the lens is behaving exactly as the lens equation says, and the image is simply being intercepted 10 cm early.
2. The student argues that because the blur shrank in proportion to the aperture, the lens must suffer from spherical aberration. Evaluate this argument and state the fix.
Read the worked answer
The scaling law is the discriminator:
- Defocus blur: b = D |delta_v|/v, so b is proportional to D (linear). Halving the aperture halves the blur.
- Spherical aberration blur: the transverse aberration of a spherical surface goes roughly as D^3, so halving the aperture cuts it by about a factor 8.
Observing 'blur halved when the aperture was halved' is the signature of defocus, not of spherical aberration, so the student's inference is backwards: her own test is evidence against her diagnosis. For a plano-convex singlet working at f/2.75, the aberration blur is only of order a fraction of a millimetre, well below 3 mm.
Depth of focus: for an acceptable blur of 0.5 mm,
delta_v = b v/D = 0.5 x 1100/40 = 14 mm.
So the screen must sit within about plus or minus 1.4 cm of 110 cm. The 10 cm error is nearly an order of magnitude outside that window, and no amount of stopping down or better figuring can bring the image into focus at 100 cm. The fix is geometric: move the screen to 110 cm (or move the lens), stop down only if the image is too bright or to trim residual aberration.
3. The wall bracket cannot be moved. Where must the grid be placed to get a sharp image on it, what magnification results, and how accurately must that position be set?
Read the worked answer
Require v = 100 cm with f = 10.0 cm and a real object:
1/u = 1/f - 1/v = 1/10.0 - 1/100 = 0.1 - 0.01 = 0.09 cm^-1, so u = 11.11 cm.
She must move the grid about 1.1 mm further from the lens. Magnification becomes
m = v/u = 100/11.11 = 9.0,
so the image is 10 percent smaller than requested - the price of the fixed screen.
Sensitivity: differentiating the lens equation, dv/du = (v/u)^2 = m^2 = 81. A 1 mm error in the grid position therefore throws the image plane out by 8.1 cm, which at this aperture would produce a blur of 40 x 81/1000 = 3.2 mm - the same mess she already has. The position must be set to about plus or minus 0.1 mm to hold the blur under 0.5 mm.
Practical recommendations: mount the grid on a micrometer-driven carriage rather than checking with a ruler; if that is impossible, relax the requirement - use a shorter focal length or lower magnification so that m^2 is smaller, or accept a fixed-focus design with the grid bonded at the computed distance. The key judgement is that at 10x the alignment tolerance, not the lens quality, sets the image sharpness.