The projected image does become dimmer per square millimeter on the surface it is projected on, but not dimmer per proportion of the photo. For example, one millionth of the projected photo would be made from the same amount of light no matter how far the screen was from the source.
OK, let me rephrase my original question just for the sake of clarity:
Do you think the Inverse Square Law applies to the projected cone of light the lens projects onto the camera’s image plane?
For the sake of clarity can you explain precisely how you are applying the inverse square law? What is the origin of the distance measurement?
Is this for a point source imaged by the lens? Here the intensity varies approximately inversely as the square of the
distance from the image plane, sufficiently far from the image. The approximation fails for distances less than the diameter of the Airy disk divided by the lens working numerical aperture - or the aberration-limited image diameter divided by NA if this is larger.
For an extended scene imaged by a small aperture lens, with aperture diameter much less than both image width and focal length, the intensity will decrease with
distance from the lens exit pupil. For large distances, the relationship is approximately inverse square - as in your example of a projector illuminating a wall. Again, there is a near field regime where the inverse square relationship clearly fails. For a 5 mm diameter aperture, the intensity does not drop to 1/4 if the distance from the aperture is doubled from 1 mm to 2 mm.
Are you considering the inverse square law as it applies to a fixed optical configuration, or are you adjusting focus or even focal length between comparisons? Are you only interested in the image plane intensity for different focal lengths, but the same physical aperture diameter (exit pupil)? Alternatively for different focal length but the same relative aperture or F-number?
It is straightforward to calculate the result in each case, but you need to specify the question sufficiently precisely. As others have said, it may be easier to work with conservation of energy or conservation of
étendue and
luminance, rather than the
inverse square law .
Thank you for saying that. It strikes me that a lot of these discussions would be greatly simplified by simply talking about the etendue or luminance. I'm not enough of an expert to know if that deals with all of the issues raised by JACS, especially in the diffraction limit.
But at the limit of my understanding, in for the case of lens with perfect refractions, then these conservation principles tell us really
only need to think about what happen on the subject side of the lens. Isn't that correct?
Jeff -- that's a perfectly good place to start.
For a lens with a fixed physical aperture (entrance pupil):
Incident intensity (illuminance) at entrance pupil = source luminance x
solid angle subtended by source x cosine(
angle of incidence at lens) x
aperture area
For a uniform subject and a sufficiently narrow field of view, cos(
angle of incidence) is approximately unity, and this becomes a simple multiplication. Otherwise we integrate over the field of view and over the lens entrance pupil.
In any case, for a fixed subject, aperture and field of view,
the total incident power is fixed. If there are no losses, all the light is projected onto the image plane, and from simple conservation of energy the
intensity at the sensor is inversely proportional to the square of the linear magnification, and so inversely proportional to the distance from lens rear principal point to the sensor.
This appears similar to an inverse square dependence, and seems to risk JACS' infinite intensity singularity as the focal length approaches zero. The problem lies in the assumption of fixed aperture and no losses. In practice a properly corrected lens satisfies the
Abbe sine condition, and has a spherical principal surface centred on the focus. A consequence is that the radius of the entrance pupil cannot be larger than the focal length of the lens, so we can't maintain the assumption of fixed aperture and no change in total light captured for the shortest focal lengths.
Another consequence (or independent constraint) is that the
numerical aperture must be less than unity for a lens in air (corresponding to
working f-number > 0.5).
Diffraction limits the spot size (proportional to wavelength/NA), so avoids infinite intensity at the sensor, even for a point source in the field of view.
HTH