Why can't I use a TC-17E or TC-20E on an f/4 lens?

The reason why Nikon says the TC-17E and TC-20E is incompatible with the 200-400mm, 500mm, and 600mm f/4 lenses has to do with the light getting to the autofocus system. Nikon's specifications say that the autofocus system needs f/5.6 or faster to work reliably. What happens when you're at an effective f/8 (f/4 plus two stop loss due to the TC-20E) is that the autofocus system becomes slower and unreliable. In a pinch, you can often use these TCs with a slow lens, but that works best if you revert to manual focus. Remember, too, that the viewfinder will get dimmer with the smaller maximum aperture, so you might also want to use Live View for focus confirmation if you can.

Update alert: The D4, D4s, D5, D500, D600, D610, D750, D800/D800E, D810, D7100, and D7200 have autofocus systems that only require f/8 and faster. Thus, these converters can be used on an f/4 lens with these cameras, though not with all autofocus sensors. Indeed, you need to carefully read the manuals for your gear. On the D500, for instance, you can go all the way down from 153 focus sensors available to 15 when you mount certain TCs on certain lenses. Moreover, as you near the stated limits of autofocus (e.g. f/8 with this set of cameras) you’re going to see less reliable autofocus and perhaps slower autofocus speeds with low contrast subjects, especially in low light.

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