Can spectacles converge sunlight to an extent that it burns the eyeball? I need to know whether wearing spectacles can cause optical harm.
I saw a movie where one pair of glasses was placed on table exposed to sunlight, then the sunlight converged and focused to a point and the table started burning, I couldn't believe that, but after that, I learned that fire can be produced by focusing sunlight with a magnifying glass, I was amazed, but I was worried that this may also happen to humans wearing glasses, though our eyeballs don't usually burn like that. Is that possible?
 A: First of all, that's a movie trope.  Only positive lenses, used by farsighted people, form a focussed spot at all.  The general range of focal lengths of such eyeglasses is considerably greater than the distance from lens to any part of the human eyeball, so the relative power density increase (over no eyeglasses at all) is rather small.
However, two important points. 
1)  NEVER look directly at the sun  (which fall under the general category of "don't be stupid")
2) Yes, positive lenses in eyeglasses or elsewhere can potentially focus sunlight to a density which leads to combustion.  There are documented cases of a vase or fishbowl causing a small fire in this way.  No, it's not going to happen just be leaving typical reading glasses on your desk near a window.
EDIT: adding an example.  A quick scan of online contact lens sales suggests the maximum standard farsighted power is +8 diopters, so I'll choose +5 diopters as a "normal case."  A diopter is an inverse meter, so 5 diopters means a focal length of 20 cm.  The distance from eyeglasses to the retina is on the order of  20 mm . If we then look at a purely geometric focussing cone, the retina is one-tenth the way to focus.  The relative spot diameter is thus 90% of the 'source' diameter, so the power density has increased by $(\frac{1}{0.9})^2$ , or a factor of 1.23 : 1 .  Considering the range of possible  light light levels (direct sun, hazy day, interior lamp, etc) is more like 50:1, the lens is not doing much.  
A: In bright sunny conditions, the eye's pupil shrinks to about 1mm diameter. If you look straight at the Sun with the pupil in this state at noon, when intensity is of the order of $1000{\rm W\,m^{-2}}$, your eye will therefore focus about $0.7{\rm mW}$ onto the retina. 
This is actually considerably below the amount of heat the superbly densely envasculated retina can safely dump, so, were it not for the shorter wavelengths in sunlight (which leads to sunburn of the retina from photochemical damage), the Sun would be a safe (albeit unpleasant) light source to stare at. There are exceptions to this rule - such as macular degeneration - but a healthy retina can easily cope with this much heat.
The danger from viewing the Sun or any other beam with an optical instrument is that it may couple radiation from a larger area than a 1mm diameter pupil through the eye's pupil. This is why binoculars, which can couple the sunlight incident on a several centimeter diameter aperture into the 1mm diameter pupil, are extremely dangerous to the sight if you were to look at the Sun with them.
Eyeglasses are not this kind of instrument, if you wear them at the correct distance from your eye. They are meant to add or subtract a small amount of optical power / correct optical aberration and do not concentrate a beam as a magnifying instrument does. In your example, the eyeglass was held at its focal length - which is typically tens of centimetres - from the paper. In this configuration, they are playing a very different optical role from when you wear them 5 millimetres or so from your eye's pupil. Correctly fitted eyeglasses when worn as they are meant to be therefore do not raise the risk of eye damage from the Sun.

Comments by Users and Answers
User Floris writes:

I am reluctant to agree that the sun would be safe to look at "but for the UV component". Do you have any authoritative reference for that? You really need to look at power density on the retina (take into account size of solar image on retina) and the heat conduction capability (it is not enough to say "densely envasculated" - what is the volumetric flow rate of blood per unit area?)

Actually the power density on the retina doesn't matter so much. Once the focussed spot is smaller than the heat diffusion length, it doesn't matter how small it is: from a heat transfer perspective, it simply looks like a point source. So if you could plot the peak temperature inside the retinal tissue as a function of the NA of a focussing objective, you would see it rising with increasing NA, but it would level off when the spot shrank below the heat diffusion length. 
The source for this statement is an ophthalmologist I worked with when designing an eye imaging instrument. It's fairly well known that the big danger from the sun is photochemical damage, not thermal damage. For gruesome animal experimentation, see
Ham, W. T., Mueller, H. A., Ruffulo, J. J. and Clarke, A. M., "Senstivity of the Retina to Radiation Damage as a Function of Wavelength", Photochemistry and Photobiology, 29 1979
This publication strongly bears the danger from photochemical versus thermal damage out: see for example Fig 1, which plots total energy dose damage threshold as a function of the time period which the energy is input into the animal's eyes over. The plot shows straight lines for visible wavelengths above about 550nm meaning that the eye is brooking a constant power input given by the slope. The slope corresponds to between $10^4{\rm W\,m^{-2}}$ and $10^5{\rm W\,m^{-2}}$ (depending on wavelength) i.e. between one and two orders of magnitude greater than the power input to the eye from staring at the Sun.
In stark contrast, the curves for the shorter wavelengths below 550nm, but particularly in UV, you get a "thresholding" effect where a total light energy threshold is reached and then damage happens - i.e. it's a bit like radiation poisoning insofar that you can only brook a certain number of photon-induced chemical transitions before enough bonds are broken. 
The contrast is even more stark and impressive when one considers that the lens of the eye blocks most UV. Dangerous levels of UV on the retina are truly paltry, and of miniscule power compared to what would be needed for damage by thermal overload alone.
Just as the danger of staring at the Sun (although not recommended) for short periods tends to be overestimated, so too does the danger of chronic, low level UV exposure from glare, at low latitudes, in snow and marine environments tends to be grossly underestimated and particularly for small children. The eye's lens blocks a great deal of harmful UV by the time someone reaches twenty years of age, but the lenses of very young eyes pass much shorter wavelengths.
Note that the focussed spotsize and intensity, although not a big factor for thermal damage for the reasons stated above, most certainly is a factor for photochemical damage. Each photon has a certain probability of breaking a bond. A small region, intense region means a high rate of damage.

User Chris H writes:


You're also assuming a light-adapted eye to start with

You are absolutely right that I'm assuming a light adapted eye. This is completely normal when one is outdoors. Your comment touches on part of the reason why looking at an eclipse (especially total) can be dangerous to sight: the diamond ring phase in particular can be damaging because the pupil has increased in size to about 7mm diameter owing to the twighlight. This represents a fifty fold increase in the light power entering the eye compared with a fully shrunken pupil. You can easily cop 20mW to 50mW power input into the eye in the diamond ring phase owing to an incompletely light adapted eye. 
