Does a lens with a given numerical aperture (NA) increase etendue, even if it follows a source (e.g. fiber) with lower NA? It is well known that the etendue of an optical system cannot be decreased - i.e. the brightness of a source cannot simply be increased by using lenses to try and focus something more tightly.
However, I'm interested to know if optical elements will always decrease the performance to the worst possible level - i.e. if I have a lens with numerical aperture (NA)=0.5, does this mean that following it with any element with an NA < 0.5 will result in losses (irrespective of the lens illumination source?)
Worded another way, let's say I'm trying to couple light from one multimode fiber into another. Each fibre has the same core diameter (let's assume a large multimode fibre with 100 um core, and NA=0.3). If I use lenses to collimate one fibre output and another identical lens to focus into the next fibre, will the coupling efficiency reduce if I choose lenses with NA > 0.3?
Or, will there be no loss of coupling efficiency since the source of light into the high NA lenses (NA=0.5) is a fiber with a lower NA (NA=0.3)?
Any comments or suggested references greatly welcomed, thanks.
 A: 
i.e. if I have a lens with numerical aperture (NA)=0.5, does this mean that following it with any element with an NA < 0.5 will result in losses (irrespective of the lens illumination source?)

Nope. As a simple counter-example, consider a fiber with a collimating lens with a window placed immediately after the lens. The window effectively has an NA of 0 (focal length infinity, diameter larger than beam --> f/# is infinity --> NA is 0), but you haven't lost any light.

If I use lenses to collimate one fibre output and another identical lens to focus into the next fibre, will the coupling efficiency reduce if I choose lenses with NA > 0.3?

Also no. If you use a lens with greater NA than your light, the light simply won't see that portion of the lens. Imagine a NA 0.3 lens after your fiber. If you now make the lens diameter larger (with the same radii on each side) you've increased the lens NA, but your beam is completely ignorant of it. 
Practically speaking, your lens needs to have some margin larger than your beam, otherwise you don't have room for mechanics or edge defects.
