Focusing white LED light into a spot as small as possible An aperture (about 1mm) is placed 2cm in fount of the LED, collimated by a f=50mm lens, then focused by a f=100mm lens into a 3x3mm spot.
The focal distance of the second lens cannot be decreased.
The lenses are simple plano-convex N-BK7, so there is spherical aberration.


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*Is it possible to decrease the spot size further without decreasing light intensity (without closing down the aperture, moving aperture away from LED)? 

*Will using a lens with larger f (say f=75mm) for collimation have a significant effect (since bigger the collimated beam, smaller the focused dot) ?
 A: Consider the lens system as imaging the aperture onto a spot. You can calculate the size of the image from the focal lengths and placement of the lenses, using the Lensmaker's Formula.  As long as all of the light that makes it through the aperture reaches the lenses, the intensity of the light in the image of the aperture is simply the intensity of light at the aperture, times the ratio of the area of the aperture to the area of the image. If you either do the algebra or try all the extreme combinations, you will find that the maximum intensity you can possibly obtain in the image is the intensity of the source LED; and you can only obtain that intensity if your lens system is big enough to gather light from all angles into which the LED emits light.
A: As S. Mcgrew stated, consider this as an imaging problem. If you collimate with an f=50 lens, and focus that with a f=100 mm lens, the nominal magnification is 2 (well, -2 if you were keeping track of signs).  Hence a 1 mm aperture is imaged to a 2 mm image size.  Some aberrations will increase that slightly.
   If you increase focal length of the first lens to 75 mm, the magnication is now -(100/75) = 1.3, so the image will be 1.3 mm in diameter. (As images get smaller, aberrations may be a proportionately larger share of the actual spot size). 
One warning - you state the 1 mm aperture is 2 cm in front of the LED.  This means that only light that goes through the apertures is already traveling close to the axis joining the LED and the aperture.  Most people would change this: they would put the LED right up to the aperture.  If you do this, you will definitely get more light through the aperture.   The configuration of your imaging lenses (focal lengths, distances, and apertures) will determine how much of that light gets to the image plane. 
Second warning - you can't have a long distance between the two lenses, depending on their aperture.  That will lead to vignetting.
The irradiance (power per unit area) at the image plane may not go up even if the spot gets smaller, it depends on how everything is set up (including lens apertures).    
