f-stop is not changed with 1.6x crop, because camera engineers
decided this would not be a good thing to happen (its just a
definition of ISO). This is why smaller sensors show more noise for
the same FOV.
No, go back to the definition of f-stop, the focal length divided
by the aperture. With a TC you are increasing the focal length but
not the aperture, so f-stop is reduced.
You are right about the definition of f-stop, and your conclusions (you did mean f-stop is increased). My explanation of the TC was not formula based, but rather showing the physics behind them, and is equally valid: the TC is projecting the lens' light onto a larger diameter circle, and the "full"-sized sensor is now smaller than the circle. There is light outside the sensor area which is lost. The light density decreases with the TC. The definition of the f-stop is such that the density is inverse proportional to the square of the f-stop, so with decreasing density, f-stop increases.
Cropping the image at the
focal plane does nothing to the lens. You said yourself the light
density doesn't change (photons per square mm).
Yes, we are in total agreement.
Smaller sensors
usually have more noise because the size of the pixels is smaller,
with less light gathering area per pixel you need to increase the
amplification of the signal more than you would if you had larger
pixels that could gather more light.
OK, now I need some help from formulas. Lets collect the facts we have.
Light density only depends on the f-number, not on the sensor size.
Smaller pixels mean less photons per pixel for the same light density.
The same f-number means less photons per pixel and time for a smaller sensor with the same number of pixels.
Take a full-frame sensor, 100mm/1.8 lens, and say for a given illumination and exposure time there are N photons collected per pixel.
With a 1.4xTC this is effectively a 140mm/2.5 lens, and each pixel has N/1.4 photons collected.
The same lens, on a 1.4x cropped sensor, without TC:
This is effectively a 140mm/1.8 lens, and each pixel has N/1.4 photons collected because their size is 1/1.4 of the original size.
So, although the f-stop is the same as for the full-frame sensor without TC, the cropped sensor has less photons collected, and this has to be compensated by the camera electronics, which will result in higher noise. This is what I wanted to say with the camera engineers and the crop factor, but it turned out to be misleading. In order for the f-stop definition to still be valid, camera electronics has to be adjusted.
In comparison with the TC/full-frame combo, the cropped sensor collects the same amount of light per pixel! No difference in noise here, because for the full-frame sensor you have to double the ISO to have the same exposure time.
DOF is an interesting point. I don't see why it should be different
between 1.0x/TC and 1.6x. Could be an interesting experiment.
Anyone with two DSLRs with different crop factors, a good tele lens
and a TC that exactly matches the cropping factor difference of the
cameras, willing to make the test?
If both cameras have the same number of pixels the DOF will be
different because if you make the same size print from both cameras
the amount of enlargment is different, the circle of confusion
changes with sensor size.
The circle of confusion and DOF are different things.
The first one refers to diffraction effects on the aperture -- they reduce the resolution and get worse for smaller apertures (larger f-numbers). A camera with crop-factor has smaller pixels, and therefore better resolves the circle of confusion as the full-frame sensor does. Cropped sensors do not allow to stop-down as much as full-frame sensor cameras because of this limitation.
DOF is the range of distances that are in focus. DOF increases with larger f-numbers. Take again the 100mm/1.8 example, used at full aperture. On the full-frame sensor without TC, we take the portrait of a face that nearly completely covers the frame (Shot 1). Something 1m behind the person is nicely out of focus. Placing the TC, we now have a smaller FOV and have to go further away to take the image frame. Now the distance ratio camera-object/object-background is increased, plus we have a smaller f-number, so the background is more in focus (Shot 2). If instead we take the 1.4x sensor camera without the TC, then FOV is the same as in Shot 2, we take the same distance, but still have f/1.8. So the cropped sensor gives a more shallow DOF, better for people photography? Perhaps for some reason we should not take into account the changed f-number in Shot 2, then Shot 2 and Shot 3 give equal DOF. I do not see why Shot 3 should give a larger DOF as Shot 2.
Your argument of the different amount of enlargement is not valid, because in one case you enlarge from a smaller sensor to the large print, and in the other you already enlarge the lens output in part in the TC, and in another part from the sensor to the print. The amount of enlargement needed from the original lens to the print is the same in both cases.
Which leads me to my original statement, which you still could not disprove, but rather gets more and more proved by calculations:
Using a 1.4x TC on a full-frame sensor has exactly the same effect as using a 1.4x crop factor sensor. (Not taking into account diffraction effects, which is valid for small f-numbers). The only difference is the magnified view in the viewfinder with the TC, for your eyes only, but not for the image.
No difference in FOV, nor in noise level, nor in DOF, nor in perspective. No difference at all. AF-speed might even be in favor of the cropped sensor, because of the smaller f-number.
Thanks for the interesting discussion (for me at least)
Achim