I won't get to technical but in short a Apocromatic is better corrected for chromatic (colour ) aberrations than an Acromatic lens.
Usually but not necessarily an Achromatic lens is 2 elements.
A three or more element is called Apocromatic in the case of the barlows your chasing.
Although as to weather it has enough colour correction to be called Apocromatic is another story.
There are also semi Apocromatic and then there are Ed lens (extra low dispersion) but what there all trying to do is focus all the spectrum to one point , i't's how well they do this as to weather it could be called Apocromatic.
There are also 4 element barlows called Powermates and technically there not a barlow but do a similar but better job than the previous but at a cost.
When light passes through a lense its split up into its RGB components (basically) When it reimerges the RGB isn't recombined back together correctly (Different frequencies or refracted to different degrees) at the focal point/plane. ie they have different focal points/planes. So when you look at something through a single lens it may appear to have a rainbow around it. Or at least red and blue halo.
An achromatic telescope has two lenses (called a doublet) the tries to refract at least two of the colours back to the same focal point/plane, red and green. So bright objects often end up with a blue halo.
The new ED (Extra low dispesion) Telescopes achieve a much better rate of success because of the types of material that is used in the lenses.
An apochromatic has three lenses that tries to bring the three colours back into focus at the same point/plane. Therefore you get no colour halo. At least in the good ones
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